Trigonometry IN CAREERS. There are many careers that use trigonometry. Several are listed below.

Transcription

1 Trigonometr. Radian and Degree Measure. Trigonometric Functions: The Unit Circle. Right Triangle Trigonometr. Trigonometric Functions of An Angle.5 Graphs of Sine and Cosine Functions.6 Graphs of Other Trigonometric Functions.7 Inverse Trigonometric Functions.8 Applications and Models In Mathematics Trigonometr is used to find relationships between the sides and angles of triangles, and to write trigonometric functions as models of real-life quantities. In Real Life Trigonometric functions are used to model quantities that are periodic. For instance, throughout the da, the depth of water at the end of a dock in Bar Harbor, Maine varies with the tides. The depth can be modeled b a trigonometric function. (See Eample 7, page 5.) Andre Jenn/Alam IN CAREERS There are man careers that use trigonometr. Several are listed below. Biologist Eercise 70, page 08 Meteorologist Eercise 99, page 8 Mechanical Engineer Eercise 95, page 9 Surveor Eercise, page 59 79

2 80 Chapter Trigonometr. RADIAN AND DEGREE MEASURE What ou should learn Describe angles. Use radian measure. Use degree measure. Use angles to model and solve real-life problems. Wh ou should learn it You can use angles to model and solve real-life problems. For instance, in Eercise 9 on page 9, ou are asked to use angles to find the speed of a biccle. Angles As derived from the Greek language, the word trigonometr means measurement of triangles. Initiall, trigonometr dealt with relationships among the sides and angles of triangles and was used in the development of astronom, navigation, and surveing. With the development of calculus and the phsical sciences in the 7th centur, a different perspective arose one that viewed the classic trigonometric relationships as functions with the set of real numbers as their domains. Consequentl, the applications of trigonometr epanded to include a vast number of phsical phenomena involving rotations and vibrations. These phenomena include sound waves, light ras, planetar orbits, vibrating strings, pendulums, and orbits of atomic particles. The approach in this tet incorporates both perspectives, starting with angles and their measure. Terminal side Terminal side Verte Initial side Initial side Wolfgang Ratta/Reuters/Corbis Angle Angle in standard position FIGURE. FIGURE. An angle is determined b rotating a ra (half-line) about its endpoint. The starting position of the ra is the initial side of the angle, and the position after rotation is the terminal side, as shown in Figure.. The endpoint of the ra is the verte of the angle. This perception of an angle fits a coordinate sstem in which the origin is the verte and the initial side coincides with the positive -ais. Such an angle is in standard position, as shown in Figure.. Positive angles are generated b counterclockwise rotation, and negative angles b clockwise rotation, as shown in Figure.. Angles are labeled with Greek letters (alpha), (beta), and (theta), as well as uppercase letters A, B, and C. In Figure., note that angles and have the same initial and terminal sides. Such angles are coterminal. Positive angle (counterclockwise) Negative angle (clockwise) α β β α FIGURE. FIGURE. Coterminal angles

3 Section. Radian and Degree Measure 8 Radian Measure r r s = r The measure of an angle is determined b the amount of rotation from the initial side to the terminal side. One wa to measure angles is in radians. This tpe of measure is especiall useful in calculus. To define a radian, ou can use a central angle of a circle, one whose verte is the center of the circle, as shown in Figure.5. Definition of Radian One radian is the measure of a central angle that intercepts an arc s equal in length to the radius r of the circle. See Figure.5. Algebraicall, this means that Arc length radius when radian FIGURE.5 s r where is measured in radians. Because the circumference of a circle is r units, it follows that a central angle of one full revolution (counterclockwise) corresponds to an arc length of radians radians r radians FIGURE.6 r r r radian r r 5 radians 6 radians s r. Moreover, because there are just over si radius lengths in a full circle, as shown in Figure.6. Because the units of measure for s and r are the same, the ratio sr has no units it is simpl a real number. Because the radian measure of an angle of one full revolution is, ou can obtain the following. revolution radians revolution 6.8, radians 6 revolution 6 radians These and other common angles are shown in Figure.7. One revolution around a circle of radius r corresponds to an angle of radians because s r r r radians. 6 FIGURE.7 Recall that the four quadrants in a coordinate sstem are numbered I, II, III, and IV. Figure.8 on page 8 shows which angles between 0 and lie in each of the four quadrants. Note that angles between 0 and are acute angles and angles between and are obtuse angles.

4 8 Chapter Trigonometr = Quadrant II < < Quadrant I 0 < < = = 0 Quadrant III Quadrant IV < < < < The phrase the terminal side of lies in a quadrant is often abbreviated b simpl saing that lies in a quadrant. The terminal sides of the quadrant angles 0,,, and do not lie within quadrants. FIGURE.8 Two angles are coterminal if the have the same initial and terminal sides. For instance, the angles 0 and are coterminal, as are the angles 6 and 6. You can find an angle that is coterminal to a given angle b adding or subtracting (one revolution), as demonstrated in Eample. A given angle has infinitel man coterminal angles. For instance, is coterminal with n 6 where n is an integer. 6 = You can review operations involving fractions in Appendi A.. Eample Sketching and Finding Coterminal Angles a. For the positive angle 6, subtract to obtain a coterminal angle 6 6. See Figure.9. b. For the positive angle, subtract to obtain a coterminal angle 5. See Figure.0. c. For the negative angle, add to obtain a coterminal angle. See Figure.. = FIGURE.9 FIGURE.0 FIGURE. Now tr Eercise 7. = 5 0 = 0

5 Section. Radian and Degree Measure 8 Two positive angles and are complementar (complements of each other) if their sum is. Two positive angles are supplementar (supplements of each other) if their sum is. See Figure.. β α β α Complementar angles FIGURE. Supplementar angles Eample Complementar and Supplementar Angles If possible, find the complement and the supplement of (a) 5 and (b) 5. a. The complement of 5 is 5 The supplement of 5 is 5 b. Because 5 is greater than, it has no complement. (Remember that complements are positive angles.) The supplement is Now tr Eercise. 90 = (60 ) 0 60 = (60 ) = (60 ) = (60 ) FIGURE. Degree Measure A second wa to measure angles is in terms of degrees, denoted b the smbol. A measure of one degree ( ) is equivalent to a rotation of 60 of a complete revolution about the verte. To measure angles, it is convenient to mark degrees on the circumference of a circle, as shown in Figure.. So, a full revolution (counterclockwise) corresponds to 60, a half revolution to 80, a quarter revolution to 90, and so on. Because radians corresponds to one complete revolution, degrees and radians are related b the equations 60 rad and From the latter equation, ou obtain 80 rad and 80 rad. rad 80 which lead to the conversion rules at the top of the net page.

6 8 Chapter Trigonometr Conversions Between Degrees and Radians rad. To convert degrees to radians, multipl degrees b To convert radians to degrees, multipl radians b rad. To appl these two conversion rules, use the basic relationship (See Figure..) rad FIGURE When no units of angle measure are specified, radian measure is implied. For instance, if ou write ou impl that radians., Eample Converting from Degrees to Radians TECHNOLOGY With calculators it is convenient to use decimal degrees to denote fractional parts of degrees. Historicall, however, fractional parts of degrees were epressed in minutes and seconds, using the prime ( ) and double prime ( ) notations, respectivel. That is, one minute 60 one second Consequentl, an angle of 6 degrees, minutes, and 7 seconds is represented b Man calculators have special kes for converting an angle in degrees, minutes, and seconds D M S to decimal degree form, and vice versa a. 5 5 deg Multipl b deg radians b deg Multipl b 80. c deg Multipl b deg radians Eample Now tr Eercise 57. Converting from Radians to Degrees a. rad rad 80 deg Multipl b 80. b. rad 9 rad 80 deg 80 Multipl b rad rad 80 deg radians rad rad 90 rad rad c. rad rad 80 deg Multipl b 80. Now tr Eercise 6. If ou have a calculator with a radian-to-degree conversion ke, tr using it to verif the result shown in part (b) of Eample.

7 Section. Radian and Degree Measure 85 Applications The radian measure formula, sr, can be used to measure arc length along a circle. s = 0 r = Arc Length For a circle of radius r, a central angle intercepts an arc of length s given b s r Length of circular arc where is measured in radians. Note that if r, then s, and the radian measure of equals the arc length. Eample 5 Finding Arc Length FIGURE.5 A circle has a radius of inches. Find the length of the arc intercepted b a central angle of 0, as shown in Figure.5. To use the formula s r, first convert 0 to radian measure. 0 0 deg 80 deg Then, using a radius of r inches, ou can find the arc length to be s r radians inches. r rad Note that the units for are determined b the units for r because is given in radian measure, which has no units. Now tr Eercise 89. Linear speed measures how fast the particle moves, and angular speed measures how fast the angle changes. B dividing the formula for arc length b t, ou can establish a relationship between linear speed v and angular speed, as shown. s r s r t t v r The formula for the length of a circular arc can be used to analze the motion of a particle moving at a constant speed along a circular path. Linear and Angular Speeds Consider a particle moving at a constant speed along a circular arc of radius r. If s is the length of the arc traveled in time t, then the linear speed v of the particle is Linear speed v Moreover, if is the angle (in radian measure) corresponding to the arc length s, then the angular speed (the lowercase Greek letter omega) of the particle is Angular speed arc length time s t. central angle time t.

8 86 Chapter Trigonometr Eample 6 Finding Linear Speed FIGURE.6 0. cm The second hand of a clock is 0. centimeters long, as shown in Figure.6. Find the linear speed of the tip of this second hand as it passes around the clock face. In one revolution, the arc length traveled is s r centimeters. Substitute for r. The time required for the second hand to travel this distance is t minute 60 seconds. So, the linear speed of the tip of the second hand is 6 ft Linear speed s t 0. centimeters 60 seconds.068 centimeters per second. Now tr Eercise. Eample 7 Finding Angular and Linear Speeds The blades of a wind turbine are 6 feet long (see Figure.7). The propeller rotates at 5 revolutions per minute. a. Find the angular speed of the propeller in radians per minute. b. Find the linear speed of the tips of the blades. FIGURE.7 a. Because each revolution generates radians, it follows that the propeller turns 5 0 radians per minute. In other words, the angular speed is Angular speed t b. The linear speed is 0 radians minute 0 radians per minute. Linear speed s t r t 60 feet minute Now tr Eercise. 0,9 feet per minute.

9 Section. Radian and Degree Measure 87 A sector of a circle is the region bounded b two radii of the circle and their intercepted arc (see Figure.8). r FIGURE.8 Area of a Sector of a Circle For a circle of radius r, the area A of a sector of the circle with central angle is given b where A r is measured in radians. Eample 8 Area of a Sector of a Circle A sprinkler on a golf course fairwa spras water over a distance of 70 feet and rotates through an angle of 0 (see Figure.9). Find the area of the fairwa watered b the sprinkler. First convert 0 to radian measure as follows. FIGURE ft 0 0 deg 80 deg radians rad Multipl b 80. Then, using and r 70, the area is A r Formula for the area of a sector of a circle square feet. Now tr Eercise 7. Substitute for r and. Simplif. Simplif.

10 88 Chapter Trigonometr. EXERCISES VOCABULARY: Fill in the blanks.. means measurement of triangles.. An is determined b rotating a ra about its endpoint.. Two angles that have the same initial and terminal sides are. See for worked-out solutions to odd-numbered eercises.. One is the measure of a central angle that intercepts an arc equal to the radius of the circle. 5. Angles that measure between 0 and are angles, and angles that measure between and are angles. 6. Two positive angles that have a sum of are angles, whereas two positive angles that have a sum of are angles. 7. The angle measure that is equivalent to a rotation of 60 of a complete revolution about an angle s verte is one degrees radians. 9. The speed of a particle is the ratio of arc length to time traveled, and the speed of a particle is the ratio of central angle to time traveled. 0. The area A of a sector of a circle with radius r and central angle, where is measured in radians, is given b the formula. SKILLS AND APPLICATIONS In Eercises 6, estimate the angle to the nearest one-half radian (a) (b) 6 6. (a) (b) 7.. In Eercises 7 0, determine two coterminal angles (one positive and one negative) for each angle. Give our answers in radians. 7. (a) (b) = = In Eercises 7, determine the quadrant in which each angle lies. (The angle measure is given in radians.) 7. (a) (b) 8. (a) (b) (a) (b) 0. (a) 5 (b) (a).5 (b).5. (a) 6.0 (b).5 In Eercises 6, sketch each angle in standard position.. (a) (b). (a) 7 (b) (a) (b) 7 = 6 9. (a) (b) 9 0. (a) (b) 0 0 = 6 5

11 Section. Radian and Degree Measure 89 In Eercises, find (if possible) the complement and supplement of each angle.. (a) (b). (a) (b). (a) (b). (a) (b).5 In Eercises 5 0, estimate the number of degrees in the angle. Use a protractor to check our answer In Eercises, determine the quadrant in which each angle lies.. (a) 0 (b) 85. (a) 8. (b) (a) 50 (b) 6. (a) 60 (b). In Eercises 5 8, sketch each angle in standard position. 5. (a) 90 (b) (a) 70 (b) 0 7. (a) 0 (b) 5 8. (a) 750 (b) 600 In Eercises 9 5, determine two coterminal angles (one positive and one negative) for each angle. Give our answers in degrees. 9. (a) 90 (b) 90 = (a) (b) = (a) (b) = = (a) (b) In Eercises 5 56, find (if possible) the complement and supplement of each angle. 5. (a) 8 (b) (a) 6 (b) (a) 50 (b) (a) 0 (b) 70 In Eercises 57 60, rewrite each angle in radian measure as a multiple of. (Do not use a calculator.) 57. (a) 0 (b) (a) 5 (b) (a) 0 (b) (a) 70 (b) In Eercises 6 6, rewrite each angle in degree measure. (Do not use a calculator.) 6. (a) (b) 6. (a) 7 (b) (a) (b) 7 6. (a) (b) In Eercises 65 7, convert the angle measure from degrees to radians. Round to three decimal places In Eercises 7 80, convert the angle measure from radians to degrees. Round to three decimal places In Eercises 8 8, convert each angle measure to decimal degree form without using a calculator. Then check our answers using a calculator. 8. (a) 5 5 (b) (a) 5 0 (b) 8. (a) (b) (a) 5 6 (b) In Eercises 85 88, convert each angle measure to degrees, minutes, and seconds without using a calculator. Then check our answers using a calculator. 85. (a) 0.6 (b) (a) 5. (b) (a).5 (b) (a) 0.6 (b) 0.79

12 90 Chapter Trigonometr In Eercises 89 9, find the length of the arc on a circle of radius r intercepted b a central angle. Radius r inches feet 9. meters 9. 0 centimeters Central Angle In Eercises 9 96, find the radian measure of the central angle of a circle of radius r that intercepts an arc of length s. Radius r Arc Length s 9. inches 8 inches 9. feet 8 feet centimeters 0.5 centimeters kilometers 50 kilometers In Eercises 97 00, use the given arc length and radius to find the angle (in radians) Cit 06. San Francisco, California Seattle, Washington Latitude N N 07. DIFFERENCE IN LATITUDES Assuming that Earth is a sphere of radius 678 kilometers, what is the difference in the latitudes of Sracuse, New York and Annapolis, Marland, where Sracuse is about 50 kilometers due north of Annapolis? 08. DIFFERENCE IN LATITUDES Assuming that Earth is a sphere of radius 678 kilometers, what is the difference in the latitudes of Lnchburg, Virginia and Mrtle Beach, South Carolina, where Lnchburg is about 00 kilometers due north of Mrtle Beach? 09. INSTRUMENTATION The pointer on a voltmeter is 6 centimeters in length (see figure). Find the angle through which the pointer rotates when it moves.5 centimeters on the scale. 0 in In Eercises 0 0, find the area of the sector of the circle with radius r and central angle. Radius r 0. 6 inches 0. millimeters 0..5 feet 0.. miles DISTANCE BETWEEN CITIES In Eercises 05 and 06, find the distance between the cities. Assume that Earth is a sphere of radius 000 miles and that the cities are on the same longitude (one cit is due north of the other). Cit 05. Dallas, Teas Omaha, Nebraska 7 Central Angle Latitude 7 9 N 5 50 N 60 6 cm FIGURE FOR 09 FIGURE FOR 0 0. ELECTRIC HOIST An electric hoist is being used to lift a beam (see figure). The diameter of the drum on the hoist is 0 inches, and the beam must be raised feet. Find the number of degrees through which the drum must rotate.. LINEAR AND ANGULAR SPEEDS A circular power saw has a 7 - inch-diameter blade that rotates at 5000 revolutions per minute. (a) Find the angular speed of the saw blade in radians per minute. (b) Find the linear speed (in feet per minute) of one of the cutting teeth as the contact the wood being cut.. LINEAR AND ANGULAR SPEEDS A carousel with a 50-foot diameter makes revolutions per minute. (a) Find the angular speed of the carousel in radians per minute. (b) Find the linear speed (in feet per minute) of the platform rim of the carousel. ft Not drawn to scale

13 Section.. LINEAR AND ANGULAR SPEEDS The diameter of a DVD is approimatel centimeters. The drive motor of the DVD plaer is controlled to rotate precisel between 00 and 500 revolutions per minute, depending on what track is being read. (a) Find an interval for the angular speed of a DVD as it rotates. (b) Find an interval for the linear speed of a point on the outermost track as the DVD rotates.. ANGULAR SPEED A two-inch-diameter pulle on an electric motor that runs at 700 revolutions per minute is connected b a belt to a four-inch-diameter pulle on a saw arbor. (a) Find the angular speed (in radians per minute) of each pulle. (b) Find the revolutions per minute of the saw. 5. ANGULAR SPEED A car is moving at a rate of 65 miles per hour, and the diameter of its wheels is feet. (a) Find the number of revolutions per minute the wheels are rotating. (b) Find the angular speed of the wheels in radians per minute. 6. ANGULAR SPEED A computerized spin balance machine rotates a 5-inch-diameter tire at 80 revolutions per minute. (a) Find the road speed (in miles per hour) at which the tire is being balanced. (b) At what rate should the spin balance machine be set so that the tire is being tested for 55 miles per hour? 7. AREA A sprinkler on a golf green is set to spra water over a distance of 5 meters and to rotate through an angle of 0. Draw a diagram that shows the region that can be irrigated with the sprinkler. Find the area of the region. 8. AREA A car s rear windshield wiper rotates 5. The total length of the wiper mechanism is 5 inches and wipes the windshield over a distance of inches. Find the area covered b the wiper. 9. SPEED OF A BICYCLE The radii of the pedal sprocket, the wheel sprocket, and the wheel of the biccle in the figure are inches, inches, and inches, respectivel. A cclist is pedaling at a rate of revolution per second. in. in. in. Radian and Degree Measure 9 (a) Find the speed of the biccle in feet per second and miles per hour. (b) Use our result from part (a) to write a function for the distance d (in miles) a cclist travels in terms of the number n of revolutions of the pedal sprocket. (c) Write a function for the distance d (in miles) a cclist travels in terms of the time t (in seconds). Compare this function with the function from part (b). (d) Classif the tpes of functions ou found in parts (b) and (c). Eplain our reasoning. 0. CAPSTONE Write a short paper in our own words eplaining the meaning of each of the following concepts to a classmate. (a) an angle in standard position (b) positive and negative angles (c) coterminal angles (d) angle measure in degrees and radians (e) obtuse and acute angles (f) complementar and supplementar angles EXPLORATION TRUE OR FALSE? In Eercises, determine whether the statement is true or false. Justif our answer.. A measurement of radians corresponds to two complete revolutions from the initial side to the terminal side of an angle.. The difference between the measures of two coterminal angles is alwas a multiple of 60 if epressed in degrees and is alwas a multiple of radians if epressed in radians.. An angle that measures 60 lies in Quadrant III.. THINK ABOUT IT A fan motor turns at a given angular speed. How does the speed of the tips of the blades change if a fan of greater diameter is installed on the motor? Eplain. 5. THINK ABOUT IT Is a degree or a radian the larger unit of measure? Eplain. 6. WRITING If the radius of a circle is increasing and the magnitude of a central angle is held constant, how is the length of the intercepted arc changing? Eplain our reasoning. 7. PROOF Prove that the area of a circular sector of radius r with central angle is A r, where is measured in radians.

14 9 Chapter Trigonometr. TRIGONOMETRIC FUNCTIONS: THE UNIT CIRCLE What ou should learn Identif a unit circle and describe its relationship to real numbers. Evaluate trigonometric functions using the unit circle. Use the domain and period to evaluate sine and cosine functions. Use a calculator to evaluate trigonometric functions. Wh ou should learn it Trigonometric functions are used to model the movement of an oscillating weight. For instance, in Eercise 60 on page 98, the displacement from equilibrium of an oscillating weight suspended b a spring is modeled as a function of time. The Unit Circle The two historical perspectives of trigonometr incorporate different methods for introducing the trigonometric functions. Our first introduction to these functions is based on the unit circle. Consider the unit circle given b as shown in Figure.0. Unit circle (, 0) (0, ) (, 0) FIGURE.0 (0, ) Richard Megna/Fundamental Photographs Imagine that the real number line is wrapped around this circle, with positive numbers corresponding to a counterclockwise wrapping and negative numbers corresponding to a clockwise wrapping, as shown in Figure.. t > 0 (, ) t (, 0) t t < 0 (, 0) t (, ) t FIGURE. As the real number line is wrapped around the unit circle, each real number t corresponds to a point, on the circle. For eample, the real number 0 corresponds to the point, 0. Moreover, because the unit circle has a circumference of, the real number also corresponds to the point, 0. In general, each real number t also corresponds to a central angle (in standard position) whose radian measure is t. With this interpretation of t, the arc length formula s r (with r ) indicates that the real number t is the (directional) length of the arc intercepted b the angle, given in radians.

15 Section. Trigonometric Functions: The Unit Circle 9 The Trigonometric Functions From the preceding discussion, it follows that the coordinates and are two functions of the real variable t. You can use these coordinates to define the si trigonometric functions of t. sine cosecant cosine secant tangent cotangent These si functions are normall abbreviated sin, csc, cos, sec, tan, and cot, respectivel. Note in the definition at the right that the functions in the second row are the reciprocals of the corresponding functions in the first row. Definitions of Trigonometric Functions Let t be a real number and let, be the point on the unit circle corresponding to t. sin t csc t, 0 cos t sec t, 0 tan t, cot t, 0 0 (0, ),, ( ), (, 0) ( ) FIGURE., (, ) (, ) (, 0) (0, ) (, ) FIGURE. ( ) (0, ) (0, ) ( ) (, 0) ( ), (, ) (, 0) ( ) (, ), (, ) In the definitions of the trigonometric functions, note that the tangent and secant are not defined when 0. For instance, because t corresponds to, 0,, it follows that tan and sec are undefined. Similarl, the cotangent and cosecant are not defined when 0. For instance, because t 0 corresponds to,, 0, cot 0 and csc 0 are undefined. In Figure., the unit circle has been divided into eight equal arcs, corresponding to t-values of Similarl, in Figure., the unit circle has been divided into equal arcs, corresponding to t-values of 0, and. 6,,,, 5 6,, 6,,,, 6, To verif the points on the unit circle in Figure., note that also lies, on the line. So, substituting for in the equation of the unit circle produces the following. 5 0,,,,,, 7 and.,, 7 5 ± Because the point is in the first quadrant, and because, ou also have You can use similar reasoning to verif the rest of the points in. Figure. and the points in Figure.. Using the, coordinates in Figures. and., ou can evaluate the trigonometric functions for common t-values. This procedure is demonstrated in Eamples,, and. You should stud and learn these eact function values for common t-values because the will help ou in later sections to perform calculations.

16 9 Chapter Trigonometr Eample Evaluating Trigonometric Functions You can review dividing fractions and rationalizing denominators in Appendi A. and Appendi A., respectivel. Evaluate the si trigonometric functions at each real number. a. t b. t 5 c. t 0 d. t 6 For each t-value, begin b finding the corresponding point, on the unit circle. Then use the definitions of trigonometric functions listed on page 9. a. t corresponds to the point 6 cos 6 tan 6 5 b. t corresponds to the point 5 sin 5 cos sin 6 5 tan,,.,,. c. t 0 corresponds to the point,, 0. csc 6 sec 6 cot 6 csc 5 sec 5 cot 5 sin 0 0 csc 0 is undefined. cos 0 sec 0 tan cot 0 is undefined. d. t corresponds to the point,, 0. sin 0 cos tan 0 0 Now tr Eercise. csc is undefined. sec cot is undefined.

17 Section. Trigonometric Functions: The Unit Circle 95 Eample Evaluating Trigonometric Functions Evaluate the si trigonometric functions at t. Moving clockwise around the unit circle, it follows that t corresponds to the point,,. sin cos tan Now tr Eercise. csc sec cot (0, ) (, 0) (, 0) (0, ) FIGURE. t =, +, +,... t = t =, +, t =, +,... t =,,... t = 0,,..., +, t =, +,... t =, +, +,... FIGURE.5 Domain and Period of Sine and Cosine The domain of the sine and cosine functions is the set of all real numbers. To determine the range of these two functions, consider the unit circle shown in Figure.. B definition, sin t and cos t. Because, is on the unit circle, ou know that and. So, the values of sine and cosine also range between and. and Adding to each value of t in the interval 0, completes a second revolution around the unit circle, as shown in Figure.5. The values of sint and cost correspond to those of sin t and cos t. Similar results can be obtained for repeated revolutions (positive or negative) on the unit circle. This leads to the general result and sin t sint n sin t cost n cos t cos t for an integer n and real number t. Functions that behave in such a repetitive (or cclic) manner are called periodic. Definition of Periodic Function A function f is periodic if there eists a positive real number c such that ft c f t for all t in the domain of f. The smallest number c for which f is periodic is called the period of f.

18 96 Chapter Trigonometr Recall from Section.5 that a function f is even if f t ft. f t f t, and is odd if Even and Odd Trigonometric Functions The cosine and secant functions are even. cost cos t sect sec t The sine, cosecant, tangent, and cotangent functions are odd. sint sin t csct csc t tant tan t cott cot t Eample Using the Period to Evaluate the Sine and Cosine From the definition of periodic function, it follows that the sine and cosine functions are periodic and have a period of. The other four trigonometric functions are also periodic, and will be discussed further in Section.6. TECHNOLOGY When evaluating trigonometric functions with a calculator, remember to enclose all fractional angle measures in parentheses. For instance, if ou want to evaluate sin t for t /6, ou should enter 6. SIN ENTER These kestrokes ield the correct value of 0.5. Note that some calculators automaticall place a left parenthesis after trigonometric functions. Check the user s guide for our calculator for specific kestrokes on how to evaluate trigonometric functions. a. Because ou have 6 6, b. Because 7 ou have, c. For sin t sint because the sine function is odd. 5, 5 Now tr Eercise 7. Evaluating Trigonometric Functions with a Calculator When evaluating a trigonometric function with a calculator, ou need to set the calculator to the desired mode of measurement (degree or radian). Most calculators do not have kes for the cosecant, secant, and cotangent functions. To evaluate these functions, ou can use the ke with their respective reciprocal functions sine, cosine, and tangent. For instance, to evaluate csc8, use the fact that csc 8 sin8 and enter the following kestroke sequence in radian mode. SIN 8 ENTER Displa.659 Eample cos 7 cos Using a Calculator Function Mode Calculator Kestrokes Displa a. sin Radian SIN ENTER b. cot.5 Radian TAN.5 ENTER Now tr Eercise 55. sin 6 sin cos 0. 6 sin 6.

19 Section. Trigonometric Functions: The Unit Circle 97. EXERCISES See for worked-out solutions to odd-numbered eercises. VOCABULARY: Fill in the blanks.. Each real number t corresponds to a point, on the.. A function f is if there eists a positive real number c such that f t c f t for all t in the domain of f.. The smallest number c for which a function f is periodic is called the of f.. A function f is if f t ftand if f t f t. SKILLS AND APPLICATIONS In Eercises 5 8, determine the eact values of the si trigonometric functions of the real number t , t ( 5, In Eercises 9 6, find the point, on the unit circle that corresponds to the real number t. 9. t 0. t. t.. t t 6. In Eercises 7 6, evaluate (if possible) the sine, cosine, and tangent of the real number t 8. t 9. t t 7. t t t t 5 t t ( 5, ( ( ( t ( ( (, t. t. t t 6. t In Eercises 7, evaluate (if possible) the si trigonometric functions of the real number. 7. t t 0.. t.. t. t In Eercises 5, evaluate the trigonometric function using its period as an aid. 5. sin 6. cos 7. cos 7 8. sin 9 9. cos 7 0. sin 9 6. sin 8. In Eercises 8, use the value of the trigonometric function to evaluate the indicated functions.. sin t. sint 8 (a) sint (a) sin t (b) csct (b) csc t 5. cost 5 6. cos t (a) cos t (a) cost (b) sect (b) sect 7. sin t 5 8. cos t 5 (a) (a) (b) sint (b) cost sin t t 5 6 t 7 t cos 9 cos t

20 98 Chapter Trigonometr In Eercises 9 58, use a calculator to evaluate the trigonometric function. Round our answer to four decimal places. (Be sure the calculator is set in the correct angle mode.) 9. sin 50. tan 5. cot 5. csc 5. cos csc sec cos sec cot HARMONIC MOTION The displacement from equilibrium of an oscillating weight suspended b a spring is given b t cos 6t, where is the displacement (in feet) and t is the time (in seconds). Find the displacements when (a) t 0, (b) t, and (c) t. 60. HARMONIC MOTION The displacement from equilibrium of an oscillating weight suspended b a spring and subject to the damping effect of friction is given b t e t cos 6t, where is the displacement (in feet) and t is the time (in seconds). (a) Complete the table. t 0 (b) Use the table feature of a graphing utilit to approimate the time when the weight reaches equilibrium. (c) What appears to happen to the displacement as t increases? EXPLORATION TRUE OR FALSE? In Eercises 6 6, determine whether the statement is true or false. Justif our answer. 6. Because sin t sin t, it can be said that the sine of a negative angle is a negative number. 6. tan a tan a 6 6. The real number 0 corresponds to the point 0, on the unit circle. 6. cos 7 cos 65. Let, and, be points on the unit circle corresponding to t t and t t, respectivel. (a) Identif the smmetr of the points, and, (b) Make a conjecture about an relationship between sin t and sin t. (c) Make a conjecture about an relationship between cos t and cos t. Use the unit circle to verif that the cosine and secant functions are even and that the sine, cosecant, tangent, and cotangent functions are odd. Verif that cos t cos t b approimating cos.5 and cos Verif that sin t t sin t sin t b approimating sin 0.5, sin 0.75, and sin. THINK ABOUT IT Because f t sin t is an odd function and g t cos t is an even function, what can be said about the function h t f t g t? THINK ABOUT IT Because f t sin t and g t tan t are odd functions, what can be said about the function h t f t g t? GRAPHICAL ANALYSIS With our graphing utilit in radian and parametric modes, enter the equations XT cos T and YT sin T and use the following settings. Tmin 0, Tma 6., Tstep 0. Xmin.5, Xma.5, Xscl Ymin, Yma, Yscl (a) Graph the entered equations and describe the graph. (b) Use the trace feature to move the cursor around the graph. What do the t-values represent? What do the - and -values represent? (c) What are the least and greatest values of and? 7. CAPSTONE A student ou are tutoring has used a unit circle divided into 8 equal parts to complete the table for selected values of t. What is wrong? t sin t 0 cos t 0 0 tan t Undef. 0 Undef. 0

21 Section. Right Triangle Trigonometr 99. RIGHT TRIANGLE TRIGONOMETRY Joseph Sohm/Visions of America/Corbis What ou should learn Evaluate trigonometric functions of acute angles. Use fundamental trigonometric identities. Use a calculator to evaluate trigonometric functions. Use trigonometric functions to model and solve real-life problems. Wh ou should learn it Trigonometric functions are often used to analze real-life situations. For instance, in Eercise 76 on page 09, ou can use trigonometric functions to find the height of a helium-filled balloon. The Si Trigonometric Functions Our second look at the trigonometric functions is from a right triangle perspective. Consider a right triangle, with one acute angle labeled, as shown in Figure.6. Relative to the angle, the three sides of the triangle are the hpotenuse, the opposite side (the side opposite the angle ), and the adjacent side (the side adjacent to the angle ). FIGURE.6 Hpotenuse Side adjacent to Using the lengths of these three sides, ou can form si ratios that define the si trigonometric functions of the acute angle. sine cosecant cosine secant tangent cotangent In the following definitions, it is important to see that 0 < < 90 lies in the first quadrant) and that for such angles the value of each trigonometric function is positive. Right Triangle Definitions of Trigonometric Functions Let be an acute angle of a right triangle. The si trigonometric functions of the angle are defined as follows. (Note that the functions in the second row are the reciprocals of the corresponding functions in the first row.) sin csc opp hp hp opp cos adj hp sec hp adj The abbreviations opp, adj, and hp represent the lengths of the three sides of a right triangle. opp the length of the side opposite adj the length of the side adjacent to hp the length of the hpotenuse Side opposite tan opp adj cot adj opp

22 00 Chapter Trigonometr Eample Evaluating Trigonometric Functions FIGURE.7 Hpotenuse You can review the Pthagorean Theorem in Section.. HISTORICAL NOTE Georg Joachim Rhaeticus (5 57) was the leading Teutonic mathematical astronomer of the 6th centur. He was the first to define the trigonometric functions as ratios of the sides of a right triangle. Use the triangle in Figure.7 to find the values of the si trigonometric functions of. B the Pthagorean Theorem, hp opp adj, it follows that hp So, the si trigonometric functions of sin 5 5. opp cos adj tan opp adj Now tr Eercise 7. are In Eample, ou were given the lengths of two sides of the right triangle, but not the angle. Often, ou will be asked to find the trigonometric functions of a given acute angle. To do this, construct a right triangle having as one of its angles. Eample hp 5 hp 5 csc sec cot hp hp adj opp 5 adj 5 opp. Evaluating Trigonometric Functions of 5 Find the values of sin 5, cos 5, and tan 5. 5 Construct a right triangle having 5 as one of its acute angles, as shown in Figure.8. Choose the length of the adjacent side to be. From geometr, ou know that the other acute angle is also 5. So, the triangle is isosceles and the length of the opposite side is also. Using the Pthagorean Theorem, ou find the length of the hpotenuse to be. 5 FIGURE.8 sin 5 opp hp cos 5 adj hp tan 5 opp adj Now tr Eercise.

23 Section. Right Triangle Trigonometr 0 Eample Evaluating Trigonometric Functions of 0 and 60 Because the angles 0, 5, and 60 6,, and occur frequentl in trigonometr, ou should learn to construct the triangles shown in Figures.8 and.9. Use the equilateral triangle shown in Figure.9 to find the values of cos 60, sin 0, and cos 0. 0 sin 60, FIGURE.9 60 TECHNOLOGY You can use a calculator to convert the answers in Eample to decimals. However, the radical form is the eact value and in most cases, the eact value is preferred. Use the Pthagorean Theorem and the equilateral triangle in Figure.9 to verif the lengths of the sides shown in the figure. For ou have adj, opp, and hp. So, sin 60 opp hp 0, and For adj, opp, and hp. So, sin 0 opp hp and Now tr Eercise 7. 60, cos 60 adj hp. cos 0 adj hp. Sines, Cosines, and Tangents of Special Angles sin 0 sin cos 0 cos tan 0 tan sin 5 sin sin 60 sin cos 5 cos cos 60 cos tan 5 tan tan 60 tan In the bo, note that sin 0 cos 60. This occurs because 0 and 60 are complementar angles. In general, it can be shown from the right triangle definitions that cofunctions of complementar angles are equal. That is, if is an acute angle, the following relationships are true. sin90 cos tan90 cot sec90 csc cos90 sin cot90 tan csc90 sec

24 0 Chapter Trigonometr Trigonometric Identities In trigonometr, a great deal of time is spent studing relationships between trigonometric functions (identities). Fundamental Trigonometric Identities Reciprocal Identities sin csc Quotient Identities Pthagorean Identities sin csc sin tan sin cos cos cos sec sec cot cos cos sin tan sec cot csc tan cot cot tan Note that sin represents sin, cos represents cos, and so on. Eample Appling Trigonometric Identities FIGURE Let be an acute angle such that sin Find the values of (a) cos and (b) tan using trigonometric identities. a. To find the value of cos, use the Pthagorean identit sin cos. So, ou have 0.6 cos Substitute 0.6 for sin. cos Subtract 0.6 from each side. cos Etract the positive square root. b. Now, knowing the sine and cosine of, ou can find the tangent of to be tan sin cos Use the definitions of cos and tan, and the triangle shown in Figure.0, to check these results. Now tr Eercise. 0.6.

25 Section. Right Triangle Trigonometr 0 Eample 5 Appling Trigonometric Identities 0 FIGURE. Let be an acute angle such that tan Find the values of (a) cot and (b) sec using trigonometric identities. a. cot Reciprocal identit tan cot b. sec tan Pthagorean identit sec sec sec 0 0 Use the definitions of cot and sec, and the triangle shown in Figure., to check these results. Now tr Eercise 5.. You can also use the reciprocal identities for sine, cosine, and tangent to evaluate the cosecant, secant, and cotangent functions with a calculator. For instance, ou could use the following kestroke sequence to evaluate sec 8. COS 8 ENTER The calculator should displa.570. Evaluating Trigonometric Functions with a Calculator To use a calculator to evaluate trigonometric functions of angles measured in degrees, first set the calculator to degree mode and then proceed as demonstrated in Section.. For instance, ou can find values of cos 8 and sec 8 as follows. Function Mode Calculator Kestrokes Displa a. cos 8 Degree COS 8 ENTER b. sec 8 Degree COS ENTER Throughout this tet, angles are assumed to be measured in radians unless noted otherwise. For eample, sin means the sine of radian and sin means the sine of degree. Eample 6 Using a Calculator Use a calculator to evaluate sec5 0. Begin b converting to decimal degree form. [Recall that Then, use a calculator to evaluate sec and Function Calculator Kestrokes Displa sec5 0 sec 5.67 Now tr Eercise 5. COS ENTER

26 0 Chapter Trigonometr Observer Observer FIGURE. Object Angle of elevation Horizontal Horizontal Angle of depression Object Applications Involving Right Triangles Man applications of trigonometr involve a process called solving right triangles. In this tpe of application, ou are usuall given one side of a right triangle and one of the acute angles and are asked to find one of the other sides, or ou are given two sides and are asked to find one of the acute angles. In Eample 7, the angle ou are given is the angle of elevation, which represents the angle from the horizontal upward to an object. For objects that lie below the horizontal, it is common to use the term angle of depression, as shown in Figure.. Eample 7 Using Trigonometr to Solve a Right Triangle A surveor is standing 5 feet from the base of the Washington Monument, as shown in Figure.. The surveor measures the angle of elevation to the top of the monument as 78.. How tall is the Washington Monument? FIGURE. = 5 ft Angle of elevation 78. Not drawn to scale From Figure., ou can see that tan 78. opp adj where 5 and is the height of the monument. So, the height of the Washington Monument is tan feet. Now tr Eercise 67. Eample 8 Using Trigonometr to Solve a Right Triangle A historic lighthouse is 00 ards from a bike path along the edge of a lake. A walkwa to the lighthouse is 00 ards long. Find the acute angle between the bike path and the walkwa, as illustrated in Figure.. 00 d 00 d FIGURE. From Figure., ou can see that the sine of the angle sin opp hp Now ou should recognize that 0. Now tr Eercise 69. is

27 Section. Right Triangle Trigonometr 05 B now ou are able to recognize that is the acute angle that satisfies the equation sin Suppose, however, that ou were given the equation sin and were asked to find the acute angle. Because sin 0 and sin ou might guess that lies somewhere between 0 and 5. In a later section, ou will stud a method b which a more precise value of can be determined. Eample 9 Solving a Right Triangle Find the length c of the skateboard ramp shown in Figure c ft FIGURE.5 From Figure.5, ou can see that sin 8. opp hp So, the length of the skateboard ramp is c c. sin feet. Now tr Eercise 7.

28 06 Chapter Trigonometr. VOCABULARY EXERCISES. Match the trigonometric function with its right triangle definition. See for worked-out solutions to odd-numbered eercises. (a) Sine (b) Cosine (c) Tangent (d) Cosecant (e) Secant (f) Cotangent hpotenuse adjacent hpotenuse adjacent opposite opposite (i) (ii) (iii) (iv) (v) (vi) adjacent opposite opposite hpotenuse hpotenuse adjacent In Eercises, fill in the blanks.. Relative to the angle, the three sides of a right triangle are the side, the side, and the.. Cofunctions of angles are equal.. An angle that measures from the horizontal upward to an object is called the angle of, whereas an angle that measures from the horizontal downward to an object is called the angle of. SKILLS AND APPLICATIONS In Eercises 5 8, find the eact values of the si trigonometric functions of the angle shown in the figure. (Use the Pthagorean Theorem to find the third side of the triangle.) In Eercises 9, find the eact values of the si trigonometric functions of the angle for each of the two triangles. Eplain wh the function values are the same In Eercises 0, sketch a right triangle corresponding to the trigonometric function of the acute angle. Use the Pthagorean Theorem to determine the third side and then find the other five trigonometric functions of.. tan. cos 5. sec 6. tan 7. sin 8. sec 9. cot 0. csc In Eercises 0, construct an appropriate triangle to complete the table. 0 90, 0 / Function (deg) (rad) Function Value. sin 0. cos 5. sec. tan 5. cot 6. csc 7. csc 5 8. sin 9. cot 0. tan

29 Section. Right Triangle Trigonometr 07 In Eercises 6, use the given function value(s), and trigonometric identities (including the cofunction identities), to find the indicated trigonometric functions.. sin 60, cos 60 (a) sin 0 (b) cos 0 (c) tan 60 (d) cot 60. sin 0 tan 0, (a) csc 0 (b) cot 60 (c) cos 0 (d) cot 0. cos (a) sin (b) tan (c) sec (d) csc90. sec (a) cos (b) cot (c) cot90 (d) sin 5. cot 5 (a) tan (b) csc (c) cot90 (d) cos 6. cos 7 (a) sec (b) sin (c) cot (d) sin90 In Eercises 7 6, use trigonometric identities to transform the left side of the equation into the right side 0 < < / tan cos tan cos sin cot sin cos sin sin cos cos cos sin. sec sec. sin cos sin 5. sin cos cos sin csc 6. 5 cot sec tan tan tan cot csc tan sec In Eercises 7 56, use a calculator to evaluate each function. Round our answers to four decimal places. (Be sure the calculator is in the correct angle mode.) 7. (a) sin 0 (b) cos (a) tan.5 (b) cot (a) sin 6.5 (b) csc (a) cot (b) sec (a) cos 50 5 (b) sec (a) sec (b) csc (a) cot 5 (b) tan 5 5. (a) sec (b) cos (a) csc 0 (b) tan (a) sec (b) cot In Eercises 57 6, find the values of in degrees 0 < < 90 and radians 0 < < / without the aid of a calculator. 57. (a) sin (b) csc 58. (a) cos (b) tan 59. (a) sec (b) cot 60. (a) tan (b) cos 6. (a) csc (b) sin 6. (a) cot (b) sec In Eercises 6 66, solve for,, or r as indicated. 6. Solve for. 6. Solve for Solve for. 66. Solve for r EMPIRE STATE BUILDING You are standing 5 meters from the base of the Empire State Building. You estimate that the angle of elevation to the top of the 86th floor (the observator) is 8. If the total height of the building is another meters above the 86th floor, what is the approimate height of the building? One of our friends is on the 86th floor. What is the distance between ou and our friend? r 0 0

30 08 Chapter Trigonometr 68. HEIGHT A si-foot person walks from the base of a broadcasting tower directl toward the tip of the shadow cast b the tower. When the person is feet from the tower and feet from the tip of the shadow, the person s shadow starts to appear beond the tower s shadow. (a) Draw a right triangle that gives a visual representation of the problem. Show the known quantities of the triangle and use a variable to indicate the height of the tower. (b) Use a trigonometric function to write an equation involving the unknown quantit. (c) What is the height of the tower? 69. ANGLE OF ELEVATION You are skiing down a mountain with a vertical height of 500 feet. The distance from the top of the mountain to the base is 000 feet. What is the angle of elevation from the base to the top of the mountain? 70. WIDTH OF A RIVER A biologist wants to know the width w of a river so that instruments for studing the pollutants in the water can be set properl. From point A, the biologist walks downstream 00 feet and sights to point C (see figure). From this sighting, it is determined that How wide is the river? 5. w C = 5 A 00 ft 7. LENGTH A gu wire runs from the ground to a cell tower. The wire is attached to the cell tower 50 feet above the ground. The angle formed between the wire and the ground is (see figure). 50 ft 7. HEIGHT OF A MOUNTAIN In traveling across flat land, ou notice a mountain directl in front of ou. Its angle of elevation (to the peak) is.5. After ou drive miles closer to the mountain, the angle of elevation is 9. Approimate the height of the mountain. 7. MACHINE SHOP CALCULATIONS A steel plate has the form of one-fourth of a circle with a radius of 60 centimeters. Two two-centimeter holes are to be drilled in the plate positioned as shown in the figure. Find the coordinates of the center of each hole (, ).5 9 mi (, ) cm FIGURE FOR 7 FIGURE FOR 7 7. MACHINE SHOP CALCULATIONS A tapered shaft has a diameter of 5 centimeters at the small end and is 5 centimeters long (see figure). The taper is. Find the diameter d of the large end of the shaft. 75. GEOMETRY Use a compass to sketch a quarter of a circle of radius 0 centimeters. Using a protractor, construct an angle of 0 in standard position (see figure). Drop a perpendicular line from the point of intersection of the terminal side of the angle and the arc of the circle. B actual measurement, calculate the coordinates, of the point of intersection and use these measurements to approimate the si trigonometric functions of a 0 angle. 0 Not drawn to scale d 5 cm = (, ) (a) How long is the gu wire? (b) How far from the base of the tower is the gu wire anchored to the ground? 0 cm 0 0

31 Section. 76. HEIGHT A 0-meter line is used to tether a helium-filled balloon. Because of a breeze, the line makes an angle of approimatel 85 with the ground. (a) Draw a right triangle that gives a visual representation of the problem. Show the known quantities of the triangle and use a variable to indicate the height of the balloon. (b) Use a trigonometric function to write an equation involving the unknown quantit. (c) What is the height of the balloon? (d) The breeze becomes stronger and the angle the balloon makes with the ground decreases. How does this affect the triangle ou drew in part (a)? (e) Complete the table, which shows the heights (in meters) of the balloon for decreasing angle measures. Angle, Height Angle, Height (f) As the angle the balloon makes with the ground approaches 0, how does this affect the height of the balloon? Draw a right triangle to eplain our reasoning. EXPLORATION 09 Right Triangle Trigonometr 8. THINK ABOUT IT (a) Complete the table sin cos (b) Discuss the behavior of the sine function for in the range from 0 to 90. (c) Discuss the behavior of the cosine function for in the range from 0 to 90. (d) Use the definitions of the sine and cosine functions to eplain the results of parts (b) and (c). 85. WRITING In right triangle trigonometr, eplain wh sin 0 regardless of the size of the triangle. 86. GEOMETRY Use the equilateral triangle shown in Figure.9 and similar triangles to verif the points in Figure. (in Section.) that do not lie on the aes. 87. THINK ABOUT IT You are given onl the value tan. Is it possible to find the value of sec without finding the measure of? Eplain. 88. CAPSTONE The Johnstown Inclined Plane in Pennslvania is one of the longest and steepest hoists in the world. The railwa cars travel a distance of feet at an angle of approimatel 5., rising to a height of 69.5 feet above sea level. TRUE OR FALSE? In Eercises 77 8, determine whether the statement is true or false. Justif our answer. 77. sin 60 csc sec 0 csc sin 5 cos cot 0 csc 0 sin sin 8. tan 5 tan 5 sin 0 8. THINK ABOUT IT (a) Complete the table sin (b) Is or sin greater for in the interval 0, 0.5? (c) As approaches 0, how do and sin compare? Eplain ft 69.5 feet above sea level 5. Not drawn to scale (a) Find the vertical rise of the inclined plane. (b) Find the elevation of the lower end of the inclined plane. (c) The cars move up the mountain at a rate of 00 feet per minute. Find the rate at which the rise verticall.

32 0 Chapter Trigonometr. TRIGONOMETRIC FUNCTIONS OF ANY ANGLE What ou should learn Evaluate trigonometric functions of an angle. Find reference angles. Evaluate trigonometric functions of real numbers. Wh ou should learn it You can use trigonometric functions to model and solve real-life problems. For instance, in Eercise 99 on page 8, ou can use trigonometric functions to model the monthl normal temperatures in New York Cit and Fairbanks, Alaska. Introduction In Section., the definitions of trigonometric functions were restricted to acute angles. In this section, the definitions are etended to cover an angle. If is an acute angle, these definitions coincide with those given in the preceding section. Definitions of Trigonometric Functions of An Angle Let be an angle in standard position with, a point on the terminal side of and r 0. sec r sin cos r (, ) tan cot, 0 r, 0, 0 csc r r, 0 Because r cannot be zero, it follows that the sine and cosine functions are defined for an real value of. However, if 0, the tangent and secant of are undefined. For eample, the tangent of 90 is undefined. Similarl, if 0, the cotangent and cosecant of are undefined. Eample Evaluating Trigonometric Functions James Urbach/SuperStock Let, be a point on the terminal side of. Find the sine, cosine, and tangent of. Referring to Figure.6, ou can see that,, and r 5 5. So, ou have the following. sin r 5 cos r 5 tan (, ) r The formula r is a result of the Distance Formula. You can review the Distance Formula in Section.. Now tr Eercise 9. FIGURE.6

33 Section. Trigonometric Functions of An Angle < < < 0 > 0 0 < < > 0 > 0 The signs of the trigonometric functions in the four quadrants can be determined from the definitions of the functions. For instance, because cos it follows that cos is positive wherever > 0, which is in Quadrants I and IV. (Remember, r is alwas positive.) In a similar manner, ou can verif the results shown in Figure.7. r, Eample Evaluating Trigonometric Functions < 0 < 0 < < > 0 < 0 < < Given tan and cos > 0, find sin and 5 sec. Note that lies in Quadrant IV because that is the onl quadrant in which the tangent is negative and the cosine is positive. Moreover, using Quadrant II sin : + cos : tan : Quadrant III sin : cos : tan : + FIGURE.7 Quadrant I sin : + cos : + tan : + Quadrant IV sin : cos : + tan : tan and the fact that is negative in Quadrant IV, ou can let 5 and. So, r 6 5 and ou have sin 5 r sec r Now tr Eercise. Eample Trigonometric Functions of Quadrant Angles (, 0) FIGURE.8 (0, ) 0 (0, ) (, 0) Evaluate the cosine and tangent functions at the four quadrant angles 0, and,,. To begin, choose a point on the terminal side of each angle, as shown in Figure.8. For each of the four points, r, and ou have the following. cos 0 r cos r 0 0 cos r cos r 0 0 tan Now tr Eercise 7. tan tan 0 undefined tan undefined,, 0, 0,,, 0, 0,

34 Chapter Trigonometr Reference Angles The values of the trigonometric functions of angles greater than 90 (or less than 0) can be determined from their values at corresponding acute angles called reference angles. Definition of Reference Angle Let be an angle in standard position. Its reference angle is the acute angle formed b the terminal side of and the horizontal ais. Figure.9 shows the reference angles for in Quadrants II, III, and IV. Quadrant II Reference angle: Reference angle: Reference angle: = (radians) = 80 (degrees) FIGURE.9 Quadrant III = (radians) = 80 (degrees) Quadrant IV = (radians) = 60 (degrees) Eample Finding Reference Angles = 00 = 60 Find the reference angle. 00. a. b. c. 5 FIGURE.0 =. FIGURE. = 5 5 FIGURE. =. 5 and 5 are coterminal. = 5 a. Because 00 lies in Quadrant IV, the angle it makes with the -ais is Degrees Figure.0 shows the angle and its reference angle b. Because. lies between.5708 and it follows that it is in Quadrant II and its reference angle is Radians Figure. shows the angle and its reference angle c. First, determine that 5 is coterminal with 5, which lies in Quadrant III. So, the reference angle is Degrees Figure. shows the angle. Now tr Eercise , and its reference angle

35 Section. Trigonometric Functions of An Angle (, ) Trigonometric Functions of Real Numbers To see how a reference angle is used to evaluate a trigonometric function, consider the point, on the terminal side of, as shown in Figure.. B definition, ou know that opp r = hp adj opp, adj FIGURE. and For the right triangle with acute angle and sides of lengths and ou have sin hp r and sin tan So, it follows that sin and sin are equal, ecept possibl in sign. The same is true for tan and tan and for the other four trigonometric functions. In all cases, the sign of the function value can be determined b the quadrant in which lies. r opp opp adj. tan., Evaluating Trigonometric Functions of An Angle To find the value of a trigonometric function of an angle :. Determine the function value for the associated reference angle.. Depending on the quadrant in which lies, affi the appropriate sign to the function value. Learning the table of values at the right is worth the effort because doing so will increase both our efficienc and our confidence. Here is a pattern for the sine function that ma help ou remember the values. sin Reverse the order to get cosine values of the same angles. B using reference angles and the special angles discussed in the preceding section, ou can greatl etend the scope of eact trigonometric values. For instance, knowing the function values of 0 means that ou know the function values of all angles for which 0 is a reference angle. For convenience, the table below shows the eact values of the trigonometric functions of special angles and quadrant angles. Trigonometric Values of Common Angles sin cos tan (degrees) (radians) Undef. 0 Undef.

36 Chapter Trigonometr Eample 5 Using Reference Angles Evaluate each trigonometric function. a. cos b. tan0 c. csc a. Because lies in Quadrant III, the reference angle is as shown in Figure 6.. Moreover, the cosine is negative in Quadrant III, so. b. Because , it follows that 0 is coterminal with the second-quadrant angle 50. So, the reference angle is , as shown in Figure.5. Finall, because the tangent is negative in Quadrant II, ou have c. Because, it follows that is coterminal with the second-quadrant angle. So, the reference angle is as shown in Figure.6. Because the cosecant is positive in Quadrant II, ou have csc csc sin. cos cos tan0 tan 0., = = = 0 = 0 = = FIGURE. FIGURE.5 FIGURE.6 Now tr Eercise 59.

37 Section. Trigonometric Functions of An Angle 5 Eample 6 Using Trigonometric Identities. Let be an angle in Quadrant II such that sin Find (a) cos and (b) tan b using trigonometric identities. a. Using the Pthagorean identit sin cos, ou obtain cos Substitute for sin. Because cos cos < 0 in Quadrant II, ou can use the negative root to obtain b. Using the trigonometric identit tan, ou obtain cos tan cos Now tr Eercise 69. sin Substitute for sin You can use a calculator to evaluate trigonometric functions, as shown in the net eample. and cos. Eample 7 Using a Calculator Use a calculator to evaluate each trigonometric function. a. cot 0 b. sin7 c. sec 9 Function Mode Calculator Kestrokes Displa a. cot 0 Degree TAN 0 ENTER b. sin7 Radian SIN 7 ENTER c. sec Radian COS ENTER Now tr Eercise 79.

38 6 Chapter Trigonometr. EXERCISES VOCABULARY: Fill in the blanks. See for worked-out solutions to odd-numbered eercises. In Eercises 6, let be an angle in standard position, with, a point on the terminal side of and r 0.. sin.. tan. sec r 7. Because r cannot be, the sine and cosine functions are for an real value of. 8. The acute positive angle that is formed b the terminal side of the angle and the horizontal ais is called the angle of and is denoted b. SKILLS AND APPLICATIONS r In Eercises 9, determine the eact values of the si trigonometric functions of the angle. 9. (a) (b) (, ) 0. (a) (b). (a) (b) (, 5) (, ). (a) (b) (, ) ( 8, 5) (, ) In Eercises 8, the point is on the terminal side of an angle in standard position. Determine the eact values of the si trigonometric functions of the angle.. 5,. 8, , 6., , 7. 8., 7 (, ) (, ) In Eercises 9, state the quadrant in which 9. sin > 0 and cos > 0 0. sin < 0 and cos < 0. sin > 0 and cos < 0. sec > 0 and cot < 0 lies. In Eercises, find the values of the si trigonometric functions of with the given constraint. Function Value Constraint. tan. cos sin tan > 0 < 0 5. sin lies in Quadrant II. 6. cos lies in Quadrant III. 7. cot 8. csc 9. sec cos cot sin > 0 < 0 < 0 0. sin sec. cot is undefined.. tan is undefined In Eercises 6, the terminal side of lies on the given line in the specified quadrant. Find the values of the si trigonometric functions of b finding a point on the line. Line Quadrant. II. III 5. 0 III 6. 0 IV

39 Section. Trigonometric Functions of An Angle 7 In Eercises 7, evaluate the trigonometric function of the quadrant angle. 7. sin 8. In Eercises 5 5, find the reference angle, and sketch and in standard position In Eercises 5 68, evaluate the sine, cosine, and tangent of the angle without using a calculator In Eercises 69 7, find the indicated trigonometric value in the specified quadrant. Function Quadrant Trigonometric Value 69. sin IV cos 70. cot II sin 7. tan III sec 7. csc IV cot 7. cos I sec 7. sec III tan csc 9. sec 0. sec. sin. cot. csc. cot In Eercises 75 90, use a calculator to evaluate the trigonometric function. Round our answer to four decimal places. (Be sure the calculator is set in the correct angle mode.) 75. sin sec cos0 78. csc0 79. tan cot sec 7 8. tan88 8. tan.5 8. cot tan 86. tan sin sec cot In Eercises 9 96, find two solutions of the equation. Give our answers in degrees 0 < 60 and in radians 0 <. Do not use a calculator. 9. (a) sin (b) sin 9. (a) cos (b) cos 9. (a) csc (b) cot 9. (a) sec (b) sec 95. (a) tan (b) cot 96. (a) sin (b) sin 97. DISTANCE An airplane, fling at an altitude of 6 miles, is on a flight path that passes directl over an observer (see figure). If is the angle of elevation from the observer to the plane, find the distance d from the observer to the plane when (a) (b) and (c) 0. d 0, 90, 6 mi Not drawn to scale 98. HARMONIC MOTION The displacement from equilibrium of an oscillating weight suspended b a spring is given b t cos 6t, where is the displacement (in centimeters) and t is the time (in seconds). Find the displacement when (a) t 0, (b) t and (c) t,. 9 csc 5

40 8 Chapter Trigonometr 99. DATA ANALYSIS: METEOROLOGY The table shows the monthl normal temperatures (in degrees Fahrenheit) for selected months in New York Cit N and Fairbanks, Alaska F. (Source: National Climatic Data Center) Month New York Cit, N Fairbanks, F Januar April Jul October December (a) Use the regression feature of a graphing utilit to find a model of the form a sin bt c d for each cit. Let t represent the month, with t corresponding to Januar. (b) Use the models from part (a) to find the monthl normal temperatures for the two cities in Februar, March, Ma, June, August, September, and November. (c) Compare the models for the two cities. 00. SALES A compan that produces snowboards, which are seasonal products, forecasts monthl sales over the net ears to be S. 0.t. cos t 6, where S is measured in thousands of units and t is the time in months, with t representing Januar 00. Predict sales for each of the following months. (a) Februar 00 (b) Februar 0 (c) June 00 (d) June 0 0. HARMONIC MOTION The displacement from equilibrium of an oscillating weight suspended b a spring and subject to the damping effect of friction is given b t e t cos 6t, where is the displacement (in centimeters) and t is the time (in seconds). Find the displacement when (a) t 0, (b) t, and (c) t. 0. ELECTRIC CIRCUITS The current I (in amperes) when 00 volts is applied to a circuit is given b I 5e t sin t, where t is the time (in seconds) after the voltage is applied. Approimate the current at t 0.7 second after the voltage is applied. EXPLORATION TRUE OR FALSE? In Eercises 0 and 0, determine whether the statement is true or false. Justif our answer. 0. In each of the four quadrants, the signs of the secant function and sine function will be the same. 0. To find the reference angle for an angle (given in degrees), find the integer n such that 0 60 n 60. The difference 60 n is the reference angle. 05. WRITING Consider an angle in standard position with r centimeters, as shown in the figure. Write a short paragraph describing the changes in the values of,, sin, cos, and tan as increases continuousl from 0 to 90. (, ) cm 06. CAPSTONE Write a short paper in our own words eplaining to a classmate how to evaluate the si trigonometric functions of an angle in standard position. Include an eplanation of reference angles and how to use them, the signs of the functions in each of the four quadrants, and the trigonometric values of common angles. Be sure to include figures or diagrams in our paper. 07. THINK ABOUT IT The figure shows point P, on a unit circle and right triangle OAP. P(, ) t r O A (a) Find sin t and cos t using the unit circle definitions of sine and cosine (from Section.). (b) What is the value of r? Eplain. (c) Use the definitions of sine and cosine given in this section to find sin and cos. Write our answers in terms of and. (d) Based on our answers to parts (a) and (c), what can ou conclude?

41 Section.5 Graphs of Sine and Cosine Functions 9.5 GRAPHS OF SINE AND COSINE FUNCTIONS What ou should learn Sketch the graphs of basic sine and cosine functions. Use amplitude and period to help sketch the graphs of sine and cosine functions. Sketch translations of the graphs of sine and cosine functions. Use sine and cosine functions to model real-life data. Wh ou should learn it Sine and cosine functions are often used in scientific calculations. For instance, in Eercise 87 on page 8, ou can use a trigonometric function to model the airflow of our respirator ccle. Basic Sine and Cosine Curves In this section, ou will stud techniques for sketching the graphs of the sine and cosine functions. The graph of the sine function is a sine curve. In Figure.7, the black portion of the graph represents one period of the function and is called one ccle of the sine curve. The gra portion of the graph indicates that the basic sine curve repeats indefinitel in the positive and negative directions. The graph of the cosine function is shown in Figure.8. Recall from Section. that the domain of the sine and cosine functions is the set of all real numbers. Moreover, the range of each function is the interval,, and each function has a period of. Do ou see how this information is consistent with the basic graphs shown in Figures.7 and.8? = sin Range: 5 Period: FIGURE.7 Karl Weatherl/Corbis Range: = cos 5 Period: FIGURE.8 Note in Figures.7 and.8 that the sine curve is smmetric with respect to the origin, whereas the cosine curve is smmetric with respect to the -ais. These properties of smmetr follow from the fact that the sine function is odd and the cosine function is even.

42 0 Chapter Trigonometr To sketch the graphs of the basic sine and cosine functions b hand, it helps to note five ke points in one period of each graph: the intercepts, maimum points, and minimum points (see Figure.9). Maimum Intercept Minimum Intercept Intercept (, ) = sin (0, 0) Quarter period Half period Period: (, 0) (, ) Three-quarter period (, 0) Full period (0, ) Intercept Minimum Maimum = cos Quarter period Period: (, 0 ) (, 0) (, ) Half period Intercept (, ) Three-quarter period Maimum Full period FIGURE.9 Eample Using Ke Points to Sketch a Sine Curve Sketch the graph of sin on the interval,. Note that sin sin indicates that the -values for the ke points will have twice the magnitude of those on the graph of sin. Divide the period into four equal parts to get the ke points for sin. Intercept Maimum Intercept Minimum Intercept 0, 0,,, B connecting these ke points with a smooth curve and etending the curve in both directions over the interval,, ou obtain the graph shown in Figure.50., 0,,, and, 0 TECHNOLOGY When using a graphing utilit to graph trigonometric functions, pa special attention to the viewing window ou use. For instance, tr graphing [sin0]/0 in the standard viewing window in radian mode. What do ou observe? Use the zoom feature to find a viewing window that displas a good view of the graph. FIGURE.50 Now tr Eercise 9. = sin 5 = sin 7

43 Section.5 Graphs of Sine and Cosine Functions Amplitude and Period In the remainder of this section ou will stud the graphic effect of each of the constants a, b, c, and d in equations of the forms d a sinb c and d a cosb c. A quick review of the transformations ou studied in Section.7 should help in this investigation. The constant factor a in a sin acts as a scaling factor a vertical stretch or vertical shrink of the basic sine curve. If the basic sine curve is stretched, and if a a >, <, the basic sine curve is shrunk. The result is that the graph of a sin ranges between a and a instead of between and. The absolute value of a is the amplitude of the function a sin. The range of the function a sin for a > 0 is a a. Definition of Amplitude of Sine and Cosine Curves The amplitude of a sin and a cos represents half the distance between the maimum and minimum values of the function and is given b Amplitude a. Eample Scaling: Vertical Shrinking and Stretching On the same coordinate aes, sketch the graph of each function. a. b. cos cos FIGURE.5 = cos = cos = cos a. Because the amplitude of is the maimum value is and the minimum value is cos,. Divide one ccle, 0, into four equal parts to get the ke points Maimum Intercept Minimum Intercept Maimum 0,, b. A similar analsis shows that the amplitude of cos is, and the ke points are and Maimum Intercept Minimum Intercept Maimum 0,,, 0,, 0,,,,, The graphs of these two functions are shown in Figure.5. Notice that the graph of cos is a vertical shrink of the graph of cos and the graph of cos is a vertical stretch of the graph of cos. Now tr Eercise., 0,, 0, and,.,.

44 Chapter Trigonometr = cos = cos You know from Section.7 that the graph of f is a reflection in the -ais of the graph of f. For instance, the graph of cos is a reflection of the graph of cos, as shown in Figure.5. Because a sin completes one ccle from 0 to, it follows that a sin b completes one ccle from 0 to b. Period of Sine and Cosine Functions Let b be a positive real number. The period of a sin b and a cos b is given b FIGURE.5 Period b. Note that if 0 < b <, the period of a sin b is greater than and represents a horizontal stretching of the graph of a sin. Similarl, if b >, the period of a sin b is less than and represents a horizontal shrinking of the graph of a sin. If b is negative, the identities sin sin and cos cos are used to rewrite the function. Eample Scaling: Horizontal Stretching Sketch the graph of sin. In general, to divide a period-interval into four equal parts, successivel add period, starting with the left endpoint of the interval. For instance, for the period-interval 6, of length, ou would successivel add 6 to get 6, 0, 6,, and as the -values for the ke points on the graph. The amplitude is. Moreover, because b, the period is Substitute for b. Now, divide the period-interval 0, into four equal parts with the values,, and to obtain the ke points on the graph. b. Intercept Maimum Intercept Minimum Intercept 0, 0,,,, 0,,, and, 0 The graph is shown in Figure.5. FIGURE.5 = sin = sin Now tr Eercise. Period:

45 Section.5 Graphs of Sine and Cosine Functions You can review the techniques for shifting, reflecting, and stretching graphs in Section.7. Translations of Sine and Cosine Curves The constant c in the general equations a sinb c and a cosb c creates a horizontal translation (shift) of the basic sine and cosine curves. Comparing a sin b with a sinb c, ou find that the graph of a sinb c completes one ccle from b c 0 to b c. B solving for, ou can find the interval for one ccle to be Left endpoint Right endpoint c b c b b. Period This implies that the period of a sinb c is b, and the graph of a sin b is shifted b an amount cb. The number cb is the phase shift. Graphs of Sine and Cosine Functions The graphs of a sinb c and a cosb c have the following characteristics. (Assume b > 0. ) Amplitude a Period b The left and right endpoints of a one-ccle interval can be determined b solving the equations b c 0 and b c. Eample Horizontal Translation Analze the graph of sin. Algebraic The amplitude is and the period is. B solving the equations and 0 ou see that the interval, 7 corresponds to one ccle of the graph. Dividing this interval into four equal parts produces the ke points Intercept Maimum Intercept Minimum Intercept, 0, 5 6,, 7, 0, Now tr Eercise 9. 6,, and 7, 0. Graphical Use a graphing utilit set in radian mode to graph sin, as shown in Figure.5. Use the minimum, maimum, and zero or root features of the graphing utilit to approimate the ke points.05, 0,.6, 0.5,.9, 0, 5.76, 0.5, and 7., 0. FIGURE.5 = sin ( 5 (

46 Chapter Trigonometr = cos( + ) Eample 5 Horizontal Translation Sketch the graph of cos. The amplitude is and the period is 0. B solving the equations Period FIGURE.55 and ou see that the interval, corresponds to one ccle of the graph. Dividing this interval into four equal parts produces the ke points Minimum Intercept Maimum Intercept Minimum,, 7, 0,,, 5, 0, and,. The graph is shown in Figure.55. Now tr Eercise 5. d The final tpe of transformation is the vertical translation caused b the constant in the equations d a sinb c and d a cosb c. The shift is d units upward for d > 0 and d units downward for d < 0. In other words, the graph oscillates about the horizontal line d instead of about the -ais. 5 = + cos Eample 6 Sketch the graph of Vertical Translation cos. The amplitude is and the period is. The ke points over the interval 0, are FIGURE.56 Period 0, 5,,,,, The graph is shown in Figure.56. Compared with the graph of f cos, the graph of cos is shifted upward two units. Now tr Eercise 57.,, and, 5.

47 Section.5 Graphs of Sine and Cosine Functions 5 Mathematical Modeling Sine and cosine functions can be used to model man real-life situations, including electric currents, musical tones, radio waves, tides, and weather patterns. Time, t Midnight A.M. A.M. 6 A.M. 8 A.M. 0 A.M. Noon Depth, Eample 7 Finding a Trigonometric Model Throughout the da, the depth of water at the end of a dock in Bar Harbor, Maine varies with the tides. The table shows the depths (in feet) at various times during the morning. (Source: Nautical Software, Inc.) a. Use a trigonometric function to model the data. b. Find the depths at 9 A.M. and P.M. c. A boat needs at least 0 feet of water to moor at the dock. During what times in the afternoon can it safel dock? Depth (in feet) FIGURE.57 Changing Tides A.M. 8 A.M. Noon Time 0 0 = 5.6 cos(0.5t.09) FIGURE.58 (.7, 0) (7., 0) = 0 t a. Begin b graphing the data, as shown in Figure.57. You can use either a sine or a cosine model. Suppose ou use a cosine model of the form a cosbt c d. The difference between the maimum height and the minimum height of the graph is twice the amplitude of the function. So, the amplitude is a maimum depth minimum depth The cosine function completes one half of a ccle between the times at which the maimum and minimum depths occur. So, the period is p time of min. depth time of ma. depth 0 which implies that b p 0.5. Because high tide occurs hours after midnight, consider the left endpoint to be cb, so c.09. Moreover, because the average depth is , it follows that d 5.7. So, ou can model the depth with the function given b 5.6 cos0.5t b. The depths at 9 A.M. and P.M. are as follows. 5.6 cos foot 5.6 cos feet 9 A.M. P.M. c. To find out when the depth is at least 0 feet, ou can graph the model with the line 0 using a graphing utilit, as shown in Figure.58. Using the intersect feature, ou can determine that the depth is at least 0 feet between : P.M. t.7 and 5:8 P.M. t 7.. Now tr Eercise

48 6 Chapter Trigonometr.5 EXERCISES See for worked-out solutions to odd-numbered eercises. VOCABULARY: Fill in the blanks.. One period of a sine or cosine function is called one of the sine or cosine curve.. The of a sine or cosine curve represents half the distance between the maimum and minimum values of the function. c. For the function given b a sinb c, represents the of the graph of the function. b. For the function given b d a cosb c, d represents a of the graph of the function. SKILLS AND APPLICATIONS In Eercises 5 8, find the period and amplitude. 5. sin 5 6. cos sin cos. sin.. sin 0. 5 sin cos cos cos sin sin cos cos In Eercises 9 6, describe the relationship between the graphs of f and g. Consider amplitude, period, and shifts. 9. f sin 0. f cos g sin g cos. f cos. f sin g cos g sin. f cos. f sin g cos g sin 5. f sin 6. f cos g sin g cos In Eercises 7 0, describe the relationship between the graphs of f and g. Consider amplitude, period, and shifts f f g g In Eercises 8, graph f and g on the same set of coordinate aes. (Include two full periods.). f sin. f sin g sin g sin. f cos. f cos g cos g cos g f g f

49 Section.5 Graphs of Sine and Cosine Functions 7 5. f 6. f sin sin g sin 7. f cos 8. f cos g cos g cos In Eercises 9 60, sketch the graph of the function. (Include two full periods.) 9. 5 sin 0. sin. cos. cos 5. cos 6. sin In Eercises 6 66, g is related to a parent function f sin or f cos. (a) Describe the sequence of transformations from f to g. (b) Sketch the graph of g. (c) Use function notation to write g in terms of f. 6. g sin 6. g sin 6. g cos 6. g cos 65. g sin 66. g sin In Eercises 67 7, use a graphing utilit to graph the function. Include two full periods. Be sure to choose an appropriate viewing window. 67. sin 68. sin 69. cos 70. cos sin 7. 0 g sin. cos. sin 57. cos 58. cos 7. sin 8. 0 cos 5. cos 5. cos 6 9. sin 50. sin t 5. sin 5. 5 cos cos cos cos6 cos sin 0t 00 GRAPHICAL REASONING In Eercises 7 76, find a and d for the function f a cos d such that the graph of f matches the figure GRAPHICAL REASONING In Eercises 77 80, find a, b, and c for the function f a sinb c such that the graph of f matches the figure f f f f In Eercises 8 and 8, use a graphing utilit to graph and in the interval [, ]. Use the graphs to find real numbers such that. 8. sin 8. cos In Eercises 8 86, write an equation for the function that is described b the given characteristics. 8. A sine curve with a period of, an amplitude of, a right phase shift of, and a vertical translation up unit 5 f f f f

50 8 Chapter Trigonometr 8. A sine curve with a period of, an amplitude of, a left phase shift of, and a vertical translation down unit 85. A cosine curve with a period of, an amplitude of, a left phase shift of, and a vertical translation down units 86. A cosine curve with a period of, an amplitude of, a right phase shift of, and a vertical translation up units 87. RESPIRATORY CYCLE For a person at rest, the velocit v (in liters per second) of airflow during a respirator ccle (the time from the beginning of one breath to the beginning of the net) is given b t v 0.85 sin where t is the time (in seconds). (Inhalation, occurs when v > 0, and ehalation occurs when v < 0. ) (a) Find the time for one full respirator ccle. (b) Find the number of ccles per minute. (c) Sketch the graph of the velocit function. 88. RESPIRATORY CYCLE After eercising for a few minutes, a person has a respirator ccle for which the velocit of airflow is approimated t b v.75 sin where t is the time (in seconds)., (Inhalation occurs when v > 0, and ehalation occurs when v < 0. ) (a) Find the time for one full respirator ccle. (b) Find the number of ccles per minute. (c) Sketch the graph of the velocit function. 89. DATA ANALYSIS: METEOROLOGY The table shows the maimum dail high temperatures in Las Vegas L and International Falls I (in degrees Fahrenheit) for month t, with t corresponding to Januar. (Source: National Climatic Data Center) Month, t Las Vegas, L International Falls, I (a) A model for the temperature in Las Vegas is given b t Lt cos Find a trigonometric model for International Falls. (b) Use a graphing utilit to graph the data points and the model for the temperatures in Las Vegas. How well does the model fit the data? (c) Use a graphing utilit to graph the data points and the model for the temperatures in International Falls. How well does the model fit the data? (d) Use the models to estimate the average maimum temperature in each cit. Which term of the models did ou use? Eplain. (e) What is the period of each model? Are the periods what ou epected? Eplain. (f) Which cit has the greater variabilit in temperature throughout the ear? Which factor of the models determines this variabilit? Eplain. 90. HEALTH The function given b P 00 0 cos 5t approimates the blood pressure P (in millimeters of mercur) at time t (in seconds) for a person at rest. (a) Find the period of the function. (b) Find the number of heartbeats per minute. 9. PIANO TUNING When tuning a piano, a technician strikes a tuning fork for the A above middle C and sets up a wave motion that can be approimated b 0.00 sin 880t, where t is the time (in seconds). (a) What is the period of the function? (b) The frequenc f is given b f p. What is the frequenc of the note? 9. DATA ANALYSIS: ASTRONOMY The percents (in decimal form) of the moon s face that was illuminated on da in the ear 009, where represents Januar, are shown in the table. (Source: U.S. Naval Observator)

51 Section.5 (a) Create a scatter plot of the data. (b) Find a trigonometric model that fits the data. (c) Add the graph of our model in part (b) to the scatter plot. How well does the model fit the data? (d) What is the period of the model? (e) Estimate the moon s percent illumination for March, FUEL CONSUMPTION The dail consumption C (in gallons) of diesel fuel on a farm is modeled b C 0..6 sin t where t is the time (in das), with t corresponding to Januar. (a) What is the period of the model? Is it what ou epected? Eplain. (b) What is the average dail fuel consumption? Which term of the model did ou use? Eplain. (c) Use a graphing utilit to graph the model. Use the graph to approimate the time of the ear when consumption eceeds 0 gallons per da. 9. FERRIS WHEEL A Ferris wheel is built such that the height h (in feet) above ground of a seat on the wheel at time t (in seconds) can be modeled b h t 5 50 sin t. 0 (a) Find the period of the model. What does the period tell ou about the ride? (b) Find the amplitude of the model. What does the amplitude tell ou about the ride? (c) Use a graphing utilit to graph one ccle of the model. EXPLORATION TRUE OR FALSE? In Eercises 95 97, determine whether the statement is true or false. Justif our answer. 95. The graph of the function given b f sin translates the graph of f sin eactl one period to the right so that the two graphs look identical. 96. The function given b cos has an amplitude that is twice that of the function given b cos. 97. The graph of cos is a reflection of the graph of sin in the -ais. 98. WRITING Sketch the graph of cos b for b,, and. How does the value of b affect the graph? How man complete ccles occur between 0 and for each value of b? Graphs of Sine and Cosine Functions WRITING Sketch the graph of sin c for c, 0, and. How does the value of c affect the graph? 00. CAPSTONE Use a graphing utilit to graph the function given b d a sin b c, for several different values of a, b, c, and d. Write a paragraph describing the changes in the graph corresponding to changes in each constant. CONJECTURE In Eercises 0 and 0, graph f and g on the same set of coordinate aes. Include two full periods. Make a conjecture about the functions. 0. f sin, g cos 0. f sin, g cos 0. Using calculus, it can be shown that the sine and cosine functions can be approimated b the polnomials sin 5 and cos! 5!!! where is in radians. (a) Use a graphing utilit to graph the sine function and its polnomial approimation in the same viewing window. How do the graphs compare? (b) Use a graphing utilit to graph the cosine function and its polnomial approimation in the same viewing window. How do the graphs compare? (c) Stud the patterns in the polnomial approimations of the sine and cosine functions and predict the net term in each. Then repeat parts (a) and (b). How did the accurac of the approimations change when an additional term was added? 0. Use the polnomial approimations of the sine and cosine functions in Eercise 0 to approimate the following function values. Compare the results with those given b a calculator. Is the error in the approimation the same in each case? Eplain. (a) sin (b) sin (c) sin 6 (d) cos 0.5 (e) cos (f) cos PROJECT: METEOROLOGY To work an etended application analzing the mean monthl temperature and mean monthl precipitation in Honolulu, Hawaii, visit this tet s website at academic.cengage.com. (Data Source: National Climatic Data Center)

52 0 Chapter Trigonometr.6 GRAPHS OF OTHER TRIGONOMETRIC FUNCTIONS What ou should learn Sketch the graphs of tangent functions. Sketch the graphs of cotangent functions. Sketch the graphs of secant and cosecant functions. Sketch the graphs of damped trigonometric functions. Wh ou should learn it Graphs of trigonometric functions can be used to model real-life situations such as the distance from a television camera to a unit in a parade, as in Eercise 9 on page 9. Graph of the Tangent Function Recall that the tangent function is odd. That is, tan tan. Consequentl, the graph of tan is smmetric with respect to the origin. You also know from the identit tan sin cos that the tangent is undefined for values at which cos 0. Two such values are ± ± tan Undef Undef. As indicated in the table, tan increases without bound as approaches from the left, and decreases without bound as approaches from the right. So, the graph of tan has vertical asmptotes at and, as shown in Figure.59. Moreover, because the period of the tangent function is, vertical asmptotes also occur when n, where n is an integer. The domain of the tangent function is the set of all real numbers other than n, and the range is the set of all real numbers Alan Pappe/Photodisc/Gett Images = tan PERIOD: DOMAIN: ALL n RANGE: (, ) VERTICAL ASYMPTOTES: SYMMETRY: ORIGIN n You can review odd and even functions in Section.5. You can review smmetr of a graph in Section.. You can review trigonometric identities in Section.. You can review asmptotes in Section.6. You can review domain and range of a function in Section.. You can review intercepts of a graph in Section.. FIGURE.59 Sketching the graph of a tanb c is similar to sketching the graph of a sinb c in that ou locate ke points that identif the intercepts and asmptotes. Two consecutive vertical asmptotes can be found b solving the equations b c and b c. The midpoint between two consecutive vertical asmptotes is an -intercept of the graph. The period of the function a tanb c is the distance between two consecutive vertical asmptotes. The amplitude of a tangent function is not defined. After plotting the asmptotes and the -intercept, plot a few additional points between the two asmptotes and sketch one ccle. Finall, sketch one or two additional ccles to the left and right.

53 Section.6 Graphs of Other Trigonometric Functions = tan Eample Sketching the Graph of a Tangent Function Sketch the graph of tan. B solving the equations and FIGURE.60 ou can see that two consecutive vertical asmptotes occur at and. Between these two asmptotes, plot a few points, including the -intercept, as shown in the table. Three ccles of the graph are shown in Figure tan Undef. 0 Undef. Now tr Eercise 5. Eample Sketching the Graph of a Tangent Function Sketch the graph of tan. 6 = tan B solving the equations and ou can see that two consecutive vertical asmptotes occur at and. Between these two asmptotes, plot a few points, including the -intercept, as shown in the table. Three ccles of the graph are shown in Figure.6. 6 FIGURE tan Undef. 0 Undef. B comparing the graphs in Eamples and, ou can see that the graph of a tanb c increases between consecutive vertical asmptotes when a > 0, and decreases between consecutive vertical asmptotes when a < 0. In other words, the graph for a < 0 is a reflection in the -ais of the graph for a > 0. Now tr Eercise 7.

54 Chapter Trigonometr TECHNOLOGY Some graphing utilities have difficult graphing trigonometric functions that have vertical asmptotes. Your graphing utilit ma connect parts of the graphs of tangent, cotangent, secant, and cosecant functions that are not supposed to be connected. To eliminate this problem, change the mode of the graphing utilit to dot mode. Graph of the Cotangent Function The graph of the cotangent function is similar to the graph of the tangent function. It also has a period of. However, from the identit cot cos sin ou can see that the cotangent function has vertical asmptotes when sin is zero, which occurs at n, where n is an integer. The graph of the cotangent function is shown in Figure.6. Note that two consecutive vertical asmptotes of the graph of a cotb c can be found b solving the equations b c 0 and b c. = cot PERIOD: DOMAIN: ALL n RANGE: (, ) n VERTICAL ASYMPTOTES: SYMMETRY: ORIGIN FIGURE.6 = cot Eample Sketching the Graph of a Cotangent Function 6 FIGURE.6 Sketch the graph of cot. B solving the equations 0 0 and ou can see that two consecutive vertical asmptotes occur at 0 and. Between these two asmptotes, plot a few points, including the -intercept, as shown in the table. Three ccles of the graph are shown in Figure.6. Note that the period is, the distance between consecutive asmptotes. 0 9 cot Undef. 0 Undef. Now tr Eercise 7.

55 Section.6 Graphs of Other Trigonometric Functions Graphs of the Reciprocal Functions The graphs of the two remaining trigonometric functions can be obtained from the graphs of the sine and cosine functions using the reciprocal identities csc sin and For instance, at a given value of, the -coordinate of sec is the reciprocal of the -coordinate of cos. Of course, when cos 0, the reciprocal does not eist. Near such values of, the behavior of the secant function is similar to that of the tangent function. In other words, the graphs of tan sin cos and have vertical asmptotes at n, where n is an integer, and the cosine is zero at these -values. Similarl, cot cos sin and sec cos. sec cos csc sin have vertical asmptotes where sin 0 that is, at n. To sketch the graph of a secant or cosecant function, ou should first make a sketch of its reciprocal function. For instance, to sketch the graph of csc, first sketch the graph of sin. Then take reciprocals of the -coordinates to obtain points on the graph of csc. This procedure is used to obtain the graphs shown in Figure.6. = csc = sec = sin = cos Sine: maimum Cosecant: relative maimum FIGURE.65 Cosecant: relative minimum Sine: minimum PERIOD: DOMAIN: ALL n RANGE: (,, ) n VERTICAL ASYMPTOTES: SYMMETRY: ORIGIN FIGURE.6 PERIOD: DOMAIN: ALL n RANGE: (,, ) VERTICAL ASYMPTOTES: n SYMMETRY: -AXIS In comparing the graphs of the cosecant and secant functions with those of the sine and cosine functions, note that the hills and valles are interchanged. For eample, a hill (or maimum point) on the sine curve corresponds to a valle (a relative minimum) on the cosecant curve, and a valle (or minimum point) on the sine curve corresponds to a hill (a relative maimum) on the cosecant curve, as shown in Figure.65. Additionall, -intercepts of the sine and cosine functions become vertical asmptotes of the cosecant and secant functions, respectivel (see Figure.65).

56 Chapter Trigonometr = csc + FIGURE.66 ( ) = sin + ( ) Eample Sketching the Graph of a Cosecant Function Sketch the graph of csc Begin b sketching the graph of sin For this function, the amplitude is and the period is. B solving the equations 0 and ou can see that one ccle of the sine function corresponds to the interval from to 7. The graph of this sine function is represented b the gra curve in Figure.66. Because the sine function is zero at the midpoint and endpoints of this interval, the corresponding cosecant function csc. sin has vertical asmptotes at,, 7, etc. The graph of the cosecant function is represented b the black curve in Figure.66. Now tr Eercise.. 7 Eample 5 Sketching the Graph of a Secant Function Sketch the graph of sec. = sec = cos Begin b sketching the graph of cos, as indicated b the gra curve in Figure.67. Then, form the graph of sec as the black curve in the figure. Note that the -intercepts of cos, 0,, 0,, 0,... correspond to the vertical asmptotes,,,... of the graph of sec. Moreover, notice that the period of cos and sec is. FIGURE.67 Now tr Eercise 5.

57 Section.6 Graphs of Other Trigonometric Functions 5 Damped Trigonometric Graphs A product of two functions can be graphed using properties of the individual functions. For instance, consider the function f sin = = as the product of the functions and sin. Using properties of absolute value and the fact that ou have 0 sin. Consequentl, which means that the graph of f sin lies between the lines and. Furthermore, because and sin, sin f sin ± at n f sin 0 at n FIGURE.68 f() = sin the graph of f touches the line or the line at n and has -intercepts at n. A sketch of f is shown in Figure.68. In the function f sin, the factor is called the damping factor. Eample 6 Damped Sine Wave Do ou see wh the graph of f sin touches the lines ± at n and wh the graph has -intercepts at n? Recall that the sine function is equal to at,, 5,... odd multiples of and is equal to 0 at,,,... multiples of. f() = e sin 6 = e = e 6 Sketch the graph of f e sin. Consider f as the product of the two functions and each of which has the set of real numbers as its domain. For an real number, ou know that e 0 and So, e sin e, which means that Furthermore, because and e e e sin e. f e sin ±e f e sin 0 sin sin. at at the graph of f touches the curves e and e at 6 n and has intercepts at n. A sketch is shown in Figure.69. Now tr Eercise n n FIGURE.69

58 6 Chapter Trigonometr Figure.70 summarizes the characteristics of the si basic trigonometric functions. = tan = sin = cos 5 DOMAIN: (, ) RANGE:, PERIOD: DOMAIN: (, ) RANGE:, PERIOD: DOMAIN: ALL n RANGE: (, ) PERIOD: = csc = sin = sec = cos = cot = tan DOMAIN: ALL n RANGE: (,, ) PERIOD: FIGURE.70 DOMAIN: ALL n RANGE: (,, ) PERIOD: DOMAIN: ALL n RANGE: (, ) PERIOD: CLASSROOM DISCUSSION Combining Trigonometric Functions Recall from Section.8 that functions can be combined arithmeticall. This also applies to trigonometric functions. For each of the functions h sin and h cos sin (a) identif two simpler functions f and g that comprise the combination, (b) use a table to show how to obtain the numerical values of h from the numerical values of f and g, and (c) use graphs of f and g to show how the graph of h ma be formed. Can ou find functions f d a sinb c and g d a cosb c such that f g 0 for all?

59 Section.6 Graphs of Other Trigonometric Functions 7.6 EXERCISES VOCABULARY: Fill in the blanks. See for worked-out solutions to odd-numbered eercises.. The tangent, cotangent, and cosecant functions are, so the graphs of these functions have smmetr with respect to the.. The graphs of the tangent, cotangent, secant, and cosecant functions all have asmptotes.. To sketch the graph of a secant or cosecant function, first make a sketch of its corresponding function.. For the functions given b f g sin, g is called the factor of the function f. 5. The period of tan is. 6. The domain of cot is all real numbers such that. 7. The range of sec is. 8. The period of csc is. SKILLS AND APPLICATIONS In Eercises 9, match the function with its graph. State the period of the function. [The graphs are labeled (a), (b), (c), (d), (e), and (f).] (a) (b) In Eercises 5 8, sketch the graph of the function. Include two full periods tan tan 7. tan 8. tan 9. sec 0. sec. csc. csc. sec. sec (c) (e) (d) (f) 5. csc 6. csc 7. cot 8. cot 9. sec 0. tan. tan. tan. csc. csc 5. sec 6. sec cot csc In Eercises 9 8, use a graphing utilit to graph the function. Include two full periods. 9. sec 0.. cot... sec tan csc sec 9. tan 0. tan. sec. sec. tan. cot 5. csc 6. sec sec 0. tan

60 8 Chapter Trigonometr In Eercises 9 56, use a graph to solve the equation on the interval [, ]. 9. tan 50. tan 5. cot 5. cot 5. sec 5. sec 55. csc 56. In Eercises 57 6, use the graph of the function to determine whether the function is even, odd, or neither. Verif our answer algebraicall. 57. f sec 58. f tan 59. g cot 60. g csc 6. f tan 6. f sec 6. g csc 6. g cot 65. GRAPHICAL REASONING Consider the functions given b and on the interval 0,. (a) Graph f and g in the same coordinate plane. (b) Approimate the interval in which f > g. (c) Describe the behavior of each of the functions as approaches. How is the behavior of g related to the behavior of f as approaches? 66. GRAPHICAL REASONING Consider the functions given b and on the interval,. (a) Use a graphing utilit to graph f and g in the same viewing window. (b) Approimate the interval in which f < g. (c) Approimate the interval in which f < g. How does the result compare with that of part (b)? Eplain. In Eercises 67 7, use a graphing utilit to graph the two equations in the same viewing window. Use the graphs to determine whether the epressions are equivalent. Verif the results algebraicall sin csc, sin sec, tan 69. f sin f tan cos sin, cot g csc g sec csc 70. tan cot, 7. cot, 7. sec, In Eercises 7 76, match the function with its graph. Describe the behavior of the function as approaches zero. [The graphs are labeled (a), (b), (c), and (d).] (a) (b) (c) CONJECTURE In Eercises 77 80, graph the functions f and g. Use the graphs to make a conjecture about the relationship between the functions f cos g sin f sin cos f sin cos cot csc tan In Eercises 8 8, use a graphing utilit to graph the function and the damping factor of the function in the same viewing window. Describe the behavior of the function as increases without bound. 8. g e sin 8. f e cos 8. f cos 8. h sin In Eercises 85 90, use a graphing utilit to graph the function. Describe the behavior of the function as approaches zero cos, > sin, > 0 (d) f sin ,, g sin f sin, g cos f cos, g cos g cos g 0

61 Section.6 Graphs of Other Trigonometric Functions g sin 88. f 9. DISTANCE A plane fling at an altitude of 7 miles above a radar antenna will pass directl over the radar antenna (see figure). Let d be the ground distance from the antenna to the point directl under the plane and let be the angle of elevation to the plane from the antenna. ( d is positive as the plane approaches the antenna.) Write d as a function of and graph the function over the interval 0 < <. d cos 89. f sin 90. h sin Not drawn to scale 7 mi 9. TELEVISION COVERAGE A television camera is on a reviewing platform 7 meters from the street on which a parade will be passing from left to right (see figure). Write the distance d from the camera to a particular unit in the parade as a function of the angle, and graph the function over the interval < <. (Consider as negative when a unit in the parade approaches from the left.) Temperature (in degrees Fahrenheit) H(t) L(t) Month of ear (a) What is the period of each function? (b) During what part of the ear is the difference between the normal high and normal low temperatures greatest? When is it smallest? (c) The sun is northernmost in the sk around June, but the graph shows the warmest temperatures at a later date. Approimate the lag time of the temperatures relative to the position of the sun. 9. SALES The projected monthl sales S (in thousands of units) of lawn mowers (a seasonal product) are modeled b S 7 t 0 cost6, where t is the time (in months), with t corresponding to Januar. Graph the sales function over ear. 95. HARMONIC MOTION An object weighing W pounds is suspended from the ceiling b a steel spring (see figure). The weight is pulled downward (positive direction) from its equilibrium position and released. The resulting motion of the weight is described b the function et cos t, t > 0, where is the distance (in feet) and t is the time (in seconds). t Not drawn to scale 7 m d Equilibrium Camera 9. METEOROLOGY The normal monthl high temperatures H (in degrees Fahrenheit) in Erie, Pennslvania are approimated b Ht cost6.58 sint6 and the normal monthl low temperatures L are approimated b Lt cost6.9 sint6 where t is the time (in months), with t corresponding to Januar (see figure). (Source: National Climatic Data Center) (a) Use a graphing utilit to graph the function. (b) Describe the behavior of the displacement function for increasing values of time t. EXPLORATION TRUE OR FALSE? In Eercises 96 and 97, determine whether the statement is true or false. Justif our answer. 96. The graph of csc can be obtained on a calculator b graphing the reciprocal of sin. 97. The graph of sec can be obtained on a calculator b graphing a translation of the reciprocal of sin.

62 0 Chapter Trigonometr 98. CAPSTONE Determine which function is represented b the graph. Do not use a calculator. Eplain our reasoning. (a) (b) f tan f tan f tan f tan f tan (i) (ii) (iii) (iv) (v) (i) (ii) (iii) (iv) (v) f f f f f sec csc csc sec csc In Eercises 99 and 00, use a graphing utilit to graph the function. Use the graph to determine the behavior of the function as c. (a) (b) ⴙ as approaches from the right ⴚ (c) ⴚ as approaches from the left ⴙ as approaches ⴚ from the right (d) ⴚ ⴚ as approaches ⴚ 99. f tan from the left As 0ⴙ, the value of f. As 0ⴚ, the value of f. As ⴙ, the value of f. As ⴚ, the value of f. 0. f cot What value does the sequence approach? 0. APPROXIMATION Using calculus, it can be shown that the tangent function can be approimated b the polnomial tan 6 5! 5! where is in radians. Use a graphing utilit to graph the tangent function and its polnomial approimation in the same viewing window. How do the graphs compare? 05. APPROXIMATION Using calculus, it can be shown that the secant function can be approimated b the polnomial sec 5!! where is in radians. Use a graphing utilit to graph the secant function and its polnomial approimation in the same viewing window. How do the graphs compare? 06. PATTERN RECOGNITION (a) Use a graphing utilit to graph each function. sin sin sin sin sin f sec In Eercises 0 and 0, use a graphing utilit to graph the function. Use the graph to determine the behavior of the function as c. (a) (b) (c) (d) (b) Starting with 0, generate a sequence,,,..., where n cos n. For eample, 0 cos 0 cos cos (b) Identif the pattern started in part (a) and find a function that continues the pattern one more term. Use a graphing utilit to graph. (c) The graphs in parts (a) and (b) approimate the periodic function in the figure. Find a function that is a better approimation. 0. f csc 0. THINK ABOUT IT Consider the function given b f cos. (a) Use a graphing utilit to graph the function and verif that there eists a zero between 0 and. Use the graph to approimate the zero.

63 Section.7 Inverse Trigonometric Functions.7 INVERSE TRIGONOMETRIC FUNCTIONS NASA What ou should learn Evaluate and graph the inverse sine function. Evaluate and graph the other inverse trigonometric functions. Evaluate and graph the compositions of trigonometric functions. Wh ou should learn it You can use inverse trigonometric functions to model and solve real-life problems. For instance, in Eercise 06 on page 9, an inverse trigonometric function can be used to model the angle of elevation from a television camera to a space shuttle launch. When evaluating the inverse sine function, it helps to remember the phrase the arcsine of is the angle (or number) whose sine is. Inverse Sine Function Recall from Section.9 that, for a function to have an inverse function, it must be one-to-one that is, it must pass the Horizontal Line Test. From Figure.7, ou can see that sin does not pass the test because different values of ield the same -value. FIGURE.7 = sin sin has an inverse function on this interval. However, if ou restrict the domain to the interval (corresponding to the black portion of the graph in Figure.7), the following properties hold.. On the interval,, the function sin is increasing.. On the interval,, sin takes on its full range of values, sin.. On the interval,, sin is one-to-one. So, on the restricted domain, sin has a unique inverse function called the inverse sine function. It is denoted b arcsin or sin. The notation sin is consistent with the inverse function notation f. The arcsin notation (read as the arcsine of ) comes from the association of a central angle with its intercepted arc length on a unit circle. So, arcsin means the angle (or arc) whose sine is. Both notations, arcsin and sin, are commonl used in mathematics, so remember that sin denotes the inverse sine function rather than sin. The values of arcsin lie in the interval arcsin. The graph of arcsin is shown in Eample. Definition of Inverse Sine Function The inverse sine function is defined b arcsin if and onl if where and. The domain of arcsin is,, and the range is,. sin

64 Chapter Trigonometr Eample Evaluating the Inverse Sine Function As with the trigonometric functions, much of the work with the inverse trigonometric functions can be done b eact calculations rather than b calculator approimations. Eact calculations help to increase our understanding of the inverse functions b relating them to the right triangle definitions of the trigonometric functions. If possible, find the eact value. a. b. sin c. sin arcsin a. Because sin for it follows that 6, arcsin Angle whose sine is b. Because sin for it follows that, 6. sin Angle whose sine is. c. It is not possible to evaluate sin when because there is no angle whose sine is. Remember that the domain of the inverse sine function is,. Now tr Eercise 5. Eample Graphing the Arcsine Function ( ), ( ), 6 ( ), FIGURE.7 (0, 0) (, ) ( ), 6 = arcsin (, ) Sketch a graph of arcsin. B definition, the equations arcsin and sin are equivalent for. So, their graphs are the same. From the interval,, ou can assign values to in the second equation to make a table of values. Then plot the points and draw a smooth curve through the points. The resulting graph for arcsin is shown in Figure.7. Note that it is the reflection (in the line ) of the black portion of the graph in Figure.7. Be sure ou see that Figure.7 shows the entire graph of the inverse sine function. Remember that the domain of arcsin is the closed interval, and the range is the closed interval,. Now tr Eercise. sin

65 Section.7 Inverse Trigonometric Functions Other Inverse Trigonometric Functions The cosine function is decreasing and one-to-one on the interval 0 in Figure.7., as shown = cos cos has an inverse function on this interval. FIGURE.7 Consequentl, on this interval the cosine function has an inverse function the inverse cosine function denoted b arccos or cos. Similarl, ou can define an inverse tangent function b restricting the domain of tan to the interval,. The following list summarizes the definitions of the three most common inverse trigonometric functions. The remaining three are defined in Eercises 5 7. Definitions of the Inverse Trigonometric Functions Function Domain Range arcsin if and onl if sin arccos if and onl if cos 0 arctan if and onl if tan < < < < The graphs of these three inverse trigonometric functions are shown in Figure.7. = arcsin = arccos = arctan, DOMAIN: RANGE:, FIGURE.7 DOMAIN:, RANGE: 0, DOMAIN: RANGE:,,

66 Chapter Trigonometr Eample Evaluating Inverse Trigonometric Functions Find the eact value. a. arccos b. c. arctan 0 d. cos tan a. Because cos, and lies in 0,, it follows that arccos Angle whose cosine is. b. Because cos, and lies in 0,, it follows that cos. Angle whose cosine is c. Because tan 0 0, and 0 lies in,, it follows that arctan 0 0. Angle whose tangent is 0 d. Because tan, and lies in,, it follows that tan Angle whose tangent is. Now tr Eercise 5. Eample Calculators and Inverse Trigonometric Functions Use a calculator to approimate the value (if possible). a. arctan8.5 b. sin 0.7 c. arccos WARNING / CAUTION Remember that the domain of the inverse sine function and the inverse cosine function is,, as indicated in Eample (c). Function Mode Calculator Kestrokes a. arctan8.5 Radian 8.5 From the displa, it follows that arctan b. sin 0.7 Radian SIN 0.7 From the displa, it follows that sin c. arccos Radian COS TAN ENTER ENTER ENTER In real number mode, the calculator should displa an error message because the domain of the inverse cosine function is,. Now tr Eercise 9. In Eample, if ou had set the calculator to degree mode, the displas would have been in degrees rather than radians. This convention is peculiar to calculators. B definition, the values of inverse trigonometric functions are alwas in radians.

67 Section.7 Inverse Trigonometric Functions 5 You can review the composition of functions in Section.8. Compositions of Functions Recall from Section.9 that for all in the domains of f and f, inverse functions have the properties f f and f f. Inverse Properties of Trigonometric Functions If and, then sinarcsin and arcsinsin. If and 0, then cosarccos and arccoscos. If is a real number and < <, then tanarctan and arctantan. Keep in mind that these inverse properties do not appl for arbitrar values of and. For instance, arcsin sin In other words, the propert arcsinsin arcsin is not valid for values of outside the interval,.. Eample 5 Using Inverse Properties If possible, find the eact value. a. tanarctan5 b. arcsin sin 5 c. coscos a. Because 5 lies in the domain of the arctan function, the inverse propert applies, and ou have tanarctan5 5. b. In this case, 5 does not lie within the range of the arcsine function,. However, 5 is coterminal with 5 which does lie in the range of the arcsine function, and ou have arcsin sin 5 arcsin sin c. The epression coscos is not defined because cos is not defined. Remember that the domain of the inverse cosine function is,. Now tr Eercise 9..

68 6 Chapter Trigonometr Eample 6 shows how to use right triangles to find eact values of compositions of inverse functions. Then, Eample 7 shows how to use right triangles to convert a trigonometric epression into an algebraic epression. This conversion technique is used frequentl in calculus. u = arccos Angle whose cosine is FIGURE.75 = 5 5 ( ) = u = arcsin ( 5 5 Angle whose sine is 5 FIGURE.76 ( Eample 6 Find the eact value. a. b. tan arccos Evaluating Compositions of Functions a. If ou let u arccos then cos u,. Because cos u is positive, u is a first-quadrant angle. You can sketch and label angle u as shown in Figure.75. Consequentl, tan arccos cos arcsin 5 tan u opp adj b. If ou let u arcsin 5, then sin u 5. Because sin u is negative, u is a fourthquadrant angle. You can sketch and label angle u as shown in Figure.76. Consequentl, cos arcsin 5 adj cos u hp 5. Now tr Eercise Eample 7 Some Problems from Calculus Write each of the following as an algebraic epression in. a. sinarccos, 0 b. cotarccos, 0 < u = arccos Angle whose cosine is FIGURE.77 () If ou let u arccos, then cos u, where. Because cos u adj hp ou can sketch a right triangle with acute angle u, as shown in Figure.77. From this triangle, ou can easil convert each epression to algebraic form. a. sinarccos sin u opp 0 hp 9, b. cotarccos cot u adj 0 < opp 9, Now tr Eercise 67. In Eample 7, similar arguments can be made for -values ling in the interval, 0.

69 Section.7 Inverse Trigonometric Functions 7.7 EXERCISES VOCABULARY: Fill in the blanks. Function Alternative Notation Domain Range. arcsin. cos. arctan. Without restrictions, no trigonometric function has a(n) function. SKILLS AND APPLICATIONS See for worked-out solutions to odd-numbered eercises. In Eercises 5 0, evaluate the epression without using a calculator. 5. arcsin 6. arcsin 0 7. arccos 8. arccos 0 9. arctan 0. arctan. cos.. arctan. arctan arccos sin arcsin 7. sin tan 0 0. cos tan In Eercises and, use a graphing utilit to graph f, g, and in the same viewing window to verif geometricall that g is the inverse function of f. (Be sure to restrict the domain of f properl.). f sin, g arcsin. f tan, g arctan In Eercises 0, use a calculator to evaluate the epression. Round our result to two decimal places.. arccos 0.7. arcsin arcsin arccos arctan 8. arctan 5 9. sin cos 0.6. arccos0.. arcsin0.5. arctan 0.9. arctan.8 5. arcsin arccos 7. tan 9 8. tan tan 7 0. tan 65 In Eercises and, determine the missing coordinates of the points on the graph of the function... = arctan In Eercises 8, use an inverse trigonometric function to write as a function of (, ) ( ),, 6 ( ) + (, ), ( ) (, ) In Eercises 9 5, use the properties of inverse trigonometric functions to evaluate the epression. 9. sinarcsin tanarctan 5 5. cosarccos0. 5. sinarcsin0. 5. arcsinsin 5. arccos cos 7 0 = arccos 6 +

70 8 Chapter Trigonometr In Eercises 55 66, find the eact value of the epression. (Hint: Sketch a right triangle.) 55. sinarctan 56. secarcsin costan 58. sin cos cosarcsin cscarctan 5 6. secarctan 5 6. tanarcsin 6. sinarccos 6. cotarctan In Eercises 67 76, write an algebraic epression that is equivalent to the epression. (Hint: Sketch a right triangle, as demonstrated in Eample 7.) In Eercises 77 and 78, use a graphing utilit to graph f and g in the same viewing window to verif that the two functions are equal. Eplain wh the are equal. Identif an asmptotes of the graphs In Eercises 79 8, fill in the blank csc cos cotarctan sinarctan cosarcsin secarctan sinarccos secarcsin tan arccos cot arctan csc arctan cos arcsin h r arctan 9 arcsin, arcsin arccos f sinarctan, f tan arccos, sec sin g g 0 6 arccos, arcsin 8. In Eercises 8 and 8, sketch a graph of the function and compare the graph of g with the graph of f arcsin In Eercises 85 90, sketch a graph of the function. 85. arccos 86. gt arccost 87. f ) arctan In Eercises 9 96, use a graphing utilit to graph the function arccos g arcsin g arcsin f arctan hv tanarccos v f arccos f arccos f arcsin f arctan f arctan f sin f cos arctan, In Eercises 97 and 98, write the function in terms of the sine function b using the identit A cos t B sin t A B sin t arctan A B. Use a graphing utilit to graph both forms of the function. What does the graph impl? 97. f t cos t sin t 98. f t cos t sin t In Eercises 99 0, fill in the blank. If not possible, state the reason. (Note: The notation c indicates that approaches c from the right and c indicates that approaches c from the left.) 99. As, the value of arcsin. 00. As, the value of arccos.

71 Section.7 Inverse Trigonometric Functions 9 0. As, the value of arctan. 0. As, the value of arcsin. 0. As, the value of arccos. 0. As, the value of arctan. 05. DOCKING A BOAT A boat is pulled in b means of a winch located on a dock 5 feet above the deck of the boat (see figure). Let be the angle of elevation from the boat to the winch and let s be the length of the rope from the winch to the boat. 5 ft (a) Write as a function of s. (b) Find when s 0 feet and s 0 feet. 06. PHOTOGRAPHY A television camera at ground level is filming the lift-off of a space shuttle at a point 750 meters from the launch pad (see figure). Let be the angle of elevation to the shuttle and let s be the height of the shuttle. (a) Write as a function of s. (b) Find when s 00 meters and s 00 meters. 07. PHOTOGRAPHY A photographer is taking a picture of a three-foot-tall painting hung in an art galler. The camera lens is foot below the lower edge of the painting (see figure). The angle subtended b the camera lens feet from the painting is arctan, s 750 m > 0. s Not drawn to scale ft ft (a) Use a graphing utilit to graph as a function of. (b) Move the cursor along the graph to approimate the distance from the picture when is maimum. (c) Identif the asmptote of the graph and discuss its meaning in the contet of the problem. 08. GRANULAR ANGLE OF REPOSE Different tpes of granular substances naturall settle at different angles when stored in cone-shaped piles. This angle is called the angle of repose (see figure). When rock salt is stored in a cone-shaped pile feet high, the diameter of the pile s base is about feet. (Source: Bulk-Store Structures, Inc.) (a) Find the angle of repose for rock salt. (b) How tall is a pile of rock salt that has a base diameter of 0 feet? 09. GRANULAR ANGLE OF REPOSE When whole corn is stored in a cone-shaped pile 0 feet high, the diameter of the pile s base is about 8 feet. (a) Find the angle of repose for whole corn. (b) How tall is a pile of corn that has a base diameter of 00 feet? 0. ANGLE OF ELEVATION An airplane flies at an altitude of 6 miles toward a point directl over an observer. Consider and as shown in the figure. 7 ft α β ft Not drawn to scale 6 mi Not drawn to scale (a) Write as a function of. (b) Find when 7 miles and mile.

72 50 Chapter Trigonometr. SECURITY PATROL A securit car with its spotlight on is parked 0 meters from a warehouse. Consider and as shown in the figure. (a) Write as a function of. (b) Find when 5 meters and meters. EXPLORATION TRUE OR FALSE? In Eercises, determine whether the statement is true or false. Justif our answer.... sin 5 6 tan 5 Not drawn to scale arctan arcsin arccos 0 m arcsin 5 6 arctan 5 5. Define the inverse cotangent function b restricting the domain of the cotangent function to the interval 0,, and sketch its graph. 6. Define the inverse secant function b restricting the domain of the secant function to the intervals 0, and,, and sketch its graph. 7. Define the inverse cosecant function b restricting the domain of the cosecant function to the intervals, 0 and 0,, and sketch its graph. 8. CAPSTONE Use the results of Eercises 5 7 to eplain how to graph (a) the inverse cotangent function, (b) the inverse secant function, and (c) the inverse cosecant function on a graphing utilit. In Eercises 9 6, use the results of Eercises 5 7 to evaluate each epression without using a calculator. 9. arcsec 0. arcsec. arccot. arccot. arccsc. arccsc 5. arccsc 6. arcsec In Eercises 7, use the results of Eercises 5 7 and a calculator to approimate the value of the epression. Round our result to two decimal places. 7. arcsec.5 8. arcsec.5 9. arccot arccot0. arccot 5. arccot 6 7. arccsc 5. arccsc 5. AREA In calculus, it is shown that the area of the region bounded b the graphs of 0,, a, and b is given b Area arctan b arctan a (see figure). Find the area for the following values of a and b. (a) a 0, b (b) a, b (c) a 0, b (d) a, b 6. THINK ABOUT IT Use a graphing utilit to graph the functions f and g 6 arctan. For > 0, it appears that g > f. Eplain wh ou know that there eists a positive real number a such that g < f for > a. Approimate the number a. 7. THINK ABOUT IT Consider the functions given b f sin and f arcsin. (a) Use a graphing utilit to graph the composite functions f f and f f. (b) Eplain wh the graphs in part (a) are not the graph of the line. Wh do the graphs of f f and f f differ? 8. PROOF Prove each identit. (a) arcsin arcsin (b) arctan arctan (c) a b arctan arctan, (d) arcsin arccos (e) arcsin arctan = + > 0

73 Section.8 Applications and Models 5.8 APPLICATIONS AND MODELS What ou should learn Solve real-life problems involving right triangles. Solve real-life problems involving directional bearings. Solve real-life problems involving harmonic motion. Wh ou should learn it Right triangles often occur in real-life situations. For instance, in Eercise 65 on page 6, right triangles are used to determine the shortest grain elevator for a grain storage bin on a farm. Applications Involving Right Triangles In this section, the three angles of a right triangle are denoted b the letters A, B, and C (where C is the right angle), and the lengths of the sides opposite these angles b the letters a, b, and c (where c is the hpotenuse). Eample Solving a Right Triangle Solve the right triangle shown in Figure.78 for all unknown sides and angles. B c a. A b = 9. C FIGURE.78 Because C 90, it follows that A B 90 and B To solve for a, use the fact that So, a 9. tan..8. Similarl, to solve for c, use the fact that So, tan A opp adj a b adj cos A hp b c c cos. Now tr Eercise 5. a b tan A. c b cos A. B c = 0 ft A 7 C b FIGURE.79 a Eample Finding a Side of a Right Triangle A safet regulation states that the maimum angle of elevation for a rescue ladder is 7. A fire department s longest ladder is 0 feet. What is the maimum safe rescue height? A sketch is shown in Figure.79. From the equation sin A ac, it follows that a c sin A 0 sin So, the maimum safe rescue height is about 0.6 feet above the height of the fire truck. Now tr Eercise 9.

74 5 Chapter Trigonometr Eample Finding a Side of a Right Triangle FIGURE.80 s a 00 ft 5 5 At a point 00 feet from the base of a building, the angle of elevation to the bottom of a smokestack is 5, whereas the angle of elevation to the top is 5, as shown in Figure.80. Find the height s of the smokestack alone. Note from Figure.80 that this problem involves two right triangles. For the smaller right triangle, use the fact that tan 5 a 00 to conclude that the height of the building is a 00 tan 5. For the larger right triangle, use the equation tan 5 a s 00 to conclude that a s 00 tan 5º. So, the height of the smokestack is s 00 tan 5 a 00 tan 5 00 tan 5 5. feet. Now tr Eercise. Eample Finding an Acute Angle of a Right Triangle A FIGURE.8 0 m Angle of depression. m.7 m A swimming pool is 0 meters long and meters wide. The bottom of the pool is slanted so that the water depth is. meters at the shallow end and meters at the deep end, as shown in Figure.8. Find the angle of depression of the bottom of the pool. Using the tangent function, ou can see that tan A opp adj So, the angle of depression is A arctan radian Now tr Eercise 9.

75 Section.8 Applications and Models 5 Trigonometr and Bearings In surveing and navigation, directions can be given in terms of bearings. A bearing measures the acute angle that a path or line of sight makes with a fied north-south line, as shown in Figure.8. For instance, the bearing S 5 E in Figure.8 means 5 degrees east of south. N 80 N N 5 W E W E W E S 5 S 5 E S N 80 W S N 5 E FIGURE.8 Eample 5 Finding Directions in Terms of Bearings A ship leaves port at noon and heads due west at 0 knots, or 0 nautical miles (nm) per hour. At P.M. the ship changes course to N 5 W, as shown in Figure.8. Find the ship s bearing and distance from the port of departure at P.M. In air navigation, bearings are measured in degrees clockwise from north. Eamples of air navigation bearings are shown below. 70 W 70 W 0 N S 80 0 N S E 90 E 90 D For triangle BCD, ou have B The two sides of this triangle can be determined to be b 0 sin 6 and For triangle ACD, ou can find angle A as follows. tan A The angle with the north-south line is So, the bearing of the ship is N 78.8 W. Finall, from triangle ACD, ou have sin A bc, which ields c b C FIGURE.8 0 nm d b d 0 0 sin cos 6 0 A arctan b 0 sin 6 sin A sin.8 5 B 57. nautical miles. Now tr Eercise 7. c d 0 cos 6. W 0 nm = (0 nm) Distance from port N S E Not drawn to scale A

76 5 Chapter Trigonometr Harmonic Motion The periodic nature of the trigonometric functions is useful for describing the motion of a point on an object that vibrates, oscillates, rotates, or is moved b wave motion. For eample, consider a ball that is bobbing up and down on the end of a spring, as shown in Figure.8. Suppose that 0 centimeters is the maimum distance the ball moves verticall upward or downward from its equilibrium (at rest) position. Suppose further that the time it takes for the ball to move from its maimum displacement above zero to its maimum displacement below zero and back again is t seconds. Assuming the ideal conditions of perfect elasticit and no friction or air resistance, the ball would continue to move up and down in a uniform and regular manner. 0 cm 0 cm 0 cm 0 cm 0 cm 0 cm 0 cm 0 cm 0 cm Equilibrium FIGURE.8 Maimum negative displacement Maimum positive displacement From this spring ou can conclude that the period (time for one complete ccle) of the motion is Period seconds its amplitude (maimum displacement from equilibrium) is Amplitude 0 centimeters and its frequenc (number of ccles per second) is Frequenc ccle per second. Motion of this nature can be described b a sine or cosine function, and is called simple harmonic motion.

77 Section.8 Applications and Models 55 Definition of Simple Harmonic Motion A point that moves on a coordinate line is said to be in simple harmonic motion if its distance d from the origin at time t is given b either d a sin t or d a cos t where a and are real numbers such that > 0. The motion has amplitude period, and frequenc. a, Eample 6 Simple Harmonic Motion Write the equation for the simple harmonic motion of the ball described in Figure.8, where the period is seconds. What is the frequenc of this harmonic motion? Because the spring is at equilibrium d 0 when t 0, ou use the equation d a sin t. Moreover, because the maimum displacement from zero is 0 and the period is, ou have Amplitude a Period Consequentl, the equation of motion is d 0 sin t. Note that the choice of a 0 or a 0 depends on whether the ball initiall moves up or down. The frequenc is Frequenc 0. FIGURE.85 FIGURE.86 ccle per second. Now tr Eercise 5. One illustration of the relationship between sine waves and harmonic motion can be seen in the wave motion resulting when a stone is dropped into a calm pool of water. The waves move outward in roughl the shape of sine (or cosine) waves, as shown in Figure.85. As an eample, suppose ou are fishing and our fishing bob is attached so that it does not move horizontall. As the waves move outward from the dropped stone, our fishing bob will move up and down in simple harmonic motion, as shown in Figure.86.

78 56 Chapter Trigonometr Eample 7 Simple Harmonic Motion Given the equation for simple harmonic motion d 6 cos t find (a) the maimum displacement, (b) the frequenc, (c) the value of d when t, and (d) the least positive value of t for which d 0. Algebraic The given equation has the form d a cos t, with a 6 and a. The maimum displacement (from the point of equilibrium) is given b the amplitude. So, the maimum displacement is 6. b. c. Frequenc d 6 cos 6 cos 6 6 ccle per unit of time d. To find the least positive value of t for which d 0, solve the equation d 6 cos t 0. First divide each side b 6 to obtain cos t 0. This equation is satisfied when. 8 t,, 5,.... Multipl these values b to obtain t,, 0,.... Graphical Use a graphing utilit set in radian mode to graph a. Use the maimum feature of the graphing utilit to estimate that the maimum displacement from the point of equilibrium 0 is 6, as shown in Figure.87. FIGURE.87 b. The period is the time for the graph to complete one ccle, which is.667. You can estimate the frequenc as follows. c. Use the trace or value feature to estimate that the value of when is 6, as shown in Figure.88. d. Use the zero or root feature to estimate that the least positive value of for which 0 is , as shown in Figure cos Frequenc 0.75 ccle per unit of time FIGURE.88 FIGURE.89 0 ( ) = 6 cos 8 = 6 cos( ) So, the least positive value of t is t. Now tr Eercise 57.

79 Section.8 Applications and Models 57.8 EXERCISES See for worked-out solutions to odd-numbered eercises. VOCABULARY: Fill in the blanks.. A measures the acute angle a path or line of sight makes with a fied north-south line.. A point that moves on a coordinate line is said to be in simple if its distance d from the origin at time t is given b either d a sin t or d a cos t.. The time for one complete ccle of a point in simple harmonic motion is its.. The number of ccles per second of a point in simple harmonic motion is its. SKILLS AND APPLICATIONS In Eercises 5, solve the right triangle shown in the figure for all unknown sides and angles. Round our answers to two decimal places. 5. A 0, b 6. B 5, c 5 7. B 7, b 8. A 8., a a, b 0. a 5, c 5. b 6, c 5. b., c A 5, B 65, c 0.5 a. B a C b A b FIGURE FOR 5 FIGURE FOR 5 8 In Eercises 5 8, find the altitude of the isosceles triangle shown in the figure. Round our answers to two decimal places. 5. b 6 6. b 0 7. b 8 8. b 5,, c 8, 7, 9. LENGTH The sun is 5 above the horizon. Find the length of a shadow cast b a building that is 00 feet tall (see figure). 0. LENGTH The sun is 0 above the horizon. Find the length of a shadow cast b a park statue that is feet tall.. HEIGHT A ladder 0 feet long leans against the side of a house. Find the height from the top of the ladder to the ground if the angle of elevation of the ladder is 80.. HEIGHT The length of a shadow of a tree is 5 feet when the angle of elevation of the sun is. Approimate the height of the tree.. HEIGHT From a point 50 feet in front of a church, the angles of elevation to the base of the steeple and the top of the steeple are 5 and 7 0, respectivel. Find the height of the steeple.. DISTANCE An observer in a lighthouse 50 feet above sea level observes two ships directl offshore. The angles of depression to the ships are and 6.5 (see figure). How far apart are the ships? 50 ft 6.5 Not drawn to scale 5. DISTANCE A passenger in an airplane at an altitude of 0 kilometers sees two towns directl to the east of the plane. The angles of depression to the towns are 8 and 55 (see figure). How far apart are the towns? ft 0 km 5 Not drawn to scale

80 58 Chapter Trigonometr 6. ALTITUDE You observe a plane approaching overhead and assume that its speed is 550 miles per hour. The angle of elevation of the plane is 6 at one time and 57 one minute later. Approimate the altitude of the plane. 7. ANGLE OF ELEVATION An engineer erects a 75-foot cellular telephone tower. Find the angle of elevation to the top of the tower at a point on level ground 50 feet from its base. 8. ANGLE OF ELEVATION The height of an outdoor basketball backboard is feet, and the backboard casts a shadow 7 feet long. (a) Draw a right triangle that gives a visual representation of the problem. Label the known and unknown quantities. (b) Use a trigonometric function to write an equation involving the unknown quantit. (c) Find the angle of elevation of the sun. 9. ANGLE OF DEPRESSION A cellular telephone tower that is 50 feet tall is placed on top of a mountain that is 00 feet above sea level. What is the angle of depression from the top of the tower to a cell phone user who is 5 horizontal miles awa and 00 feet above sea level? 0. ANGLE OF DEPRESSION A Global Positioning Sstem satellite orbits,500 miles above Earth s surface (see figure). Find the angle of depression from the satellite to the horizon. Assume the radius of Earth is 000 miles. 000 mi Not drawn to scale,500 mi Angle of depression GPS satellite (a) Find the length l of the tether ou are holding in terms of h, the height of the balloon from top to bottom. (b) Find an epression for the angle of elevation from ou to the top of the balloon. (c) Find the height h of the balloon if the angle of elevation to the top of the balloon is 5.. HEIGHT The designers of a water park are creating a new slide and have sketched some preliminar drawings. The length of the ladder is 0 feet, and its angle of elevation is 60 (see figure). 0 ft h 60 (a) Find the height h of the slide. (b) Find the angle of depression from the top of the slide to the end of the slide at the ground in terms of the horizontal distance d the rider travels. (c) The angle of depression of the ride is bounded b safet restrictions to be no less than 5 and not more than 0. Find an interval for how far the rider travels horizontall.. SPEED ENFORCEMENT A police department has set up a speed enforcement zone on a straight length of highwa. A patrol car is parked parallel to the zone, 00 feet from one end and 50 feet from the other end (see figure). d Enforcement zone. HEIGHT You are holding one of the tethers attached to the top of a giant character balloon in a parade. Before the start of the parade the balloon is upright and the bottom is floating approimatel 0 feet above ground level. You are standing approimatel 00 feet ahead of the balloon (see figure). 00 ft Not drawn to scale l A B 50 ft l ft 00 ft 0 ft h Not drawn to scale (a) Find the length l of the zone and the measures of the angles A and B (in degrees). (b) Find the minimum amount of time (in seconds) it takes for a vehicle to pass through the zone without eceeding the posted speed limit of 5 miles per hour.

81 Section.8 Applications and Models 59. AIRPLANE ASCENT During takeoff, an airplane s angle of ascent is 8 and its speed is 75 feet per second. (a) Find the plane s altitude after minute. (b) How long will it take the plane to climb to an altitude of 0,000 feet? 5. NAVIGATION An airplane fling at 600 miles per hour has a bearing of 5. After fling for.5 hours, how far north and how far east will the plane have traveled from its point of departure? 6. NAVIGATION A jet leaves Reno, Nevada and is headed toward Miami, Florida at a bearing of 00. The distance between the two cities is approimatel 7 miles. (a) How far north and how far west is Reno relative to Miami? (b) If the jet is to return directl to Reno from Miami, at what bearing should it travel? 7. NAVIGATION A ship leaves port at noon and has a bearing of S 9 W. The ship sails at 0 knots. (a) How man nautical miles south and how man nautical miles west will the ship have traveled b 6:00 P.M.? (b) At 6:00 P.M., the ship changes course to due west. Find the ship s bearing and distance from the port of departure at 7:00 P.M. 8. NAVIGATION A privatel owned acht leaves a dock in Mrtle Beach, South Carolina and heads toward Freeport in the Bahamas at a bearing of S. E. The acht averages a speed of 0 knots over the 8 nautical-mile trip. (a) How long will it take the acht to make the trip? (b) How far east and south is the acht after hours? (c) If a plane leaves Mrtle Beach to fl to Freeport, what bearing should be taken? 9. NAVIGATION A ship is 5 miles east and 0 miles south of port. The captain wants to sail directl to port. What bearing should be taken? 0. NAVIGATION An airplane is 60 miles north and 85 miles east of an airport. The pilot wants to fl directl to the airport. What bearing should be taken?. SURVEYING A surveor wants to find the distance across a swamp (see figure). The bearing from A to B is N W. The surveor walks 50 meters from A, and at the point C the bearing to B is N 68 W. Find (a) the bearing from A to C and (b) the distance from A to B. FIGURE FOR. LOCATION OF A FIRE Two fire towers are 0 kilometers apart, where tower A is due west of tower B. A fire is spotted from the towers, and the bearings from A and B are N 76 E and N 56 W, respectivel (see figure). Find the distance d of the fire from the line segment AB. GEOMETRY In Eercises and, find the angle between two nonvertical lines L and L. The angle satisfies the equation tan m m m m where m and m are the slopes of L and L, respectivel. (Assume that m m. ). L : 5. L : 8 L : L : 5 5. GEOMETRY Determine the angle between the diagonal of a cube and the diagonal of its base, as shown in the figure. a A W B N S E 76 d 56 a FIGURE FOR 5 FIGURE FOR 6 a A 0 km 50 m Not drawn to scale 6. GEOMETRY Determine the angle between the diagonal of a cube and its edge, as shown in the figure. W C N S E B a a

82 60 Chapter Trigonometr 7. GEOMETRY Find the length of the sides of a regular pentagon inscribed in a circle of radius 5 inches. 8. GEOMETRY Find the length of the sides of a regular heagon inscribed in a circle of radius 5 inches. 9. HARDWARE Write the distance across the flat sides of a heagonal nut as a function of r (see figure). FIGURE FOR 9 FIGURE FOR BOLT HOLES The figure shows a circular piece of sheet metal that has a diameter of 0 centimeters and contains equall-spaced bolt holes. Determine the straight-line distance between the centers of consecutive bolt holes. TRUSSES In Eercises 5 and 5, find the lengths of all the unknown members of the truss. 5. b a r 60 a b 0 c 6 ft 0 cm HARMONIC MOTION In Eercises 5 56, find a model for simple harmonic motion satisfing the specified conditions. 9 ft 6 ft 6 ft Displacement Amplitude Period t centimeters seconds 5. 0 meters 6 seconds 55. inches inches.5 seconds 56. feet feet 0 seconds HARMONIC MOTION In Eercises 57 60, for the simple harmonic motion described b the trigonometric function, find (a) the maimum displacement, (b) the frequenc, (c) the value of d when t 5, and (d) the least positive value of t for which d 0. Use a graphing utilit to verif our results cm 57. d 9 cos t 59. d sin 6t TUNING FORK A point on the end of a tuning fork moves in simple harmonic motion described b d a sin t. Find given that the tuning fork for middle C has a frequenc of 6 vibrations per second. 6. WAVE MOTION A buo oscillates in simple harmonic motion as waves go past. It is noted that the buo moves a total of.5 feet from its low point to its high point (see figure), and that it returns to its high point ever 0 seconds. Write an equation that describes the motion of the buo if its high point is at t 0. Equilibrium 6. OSCILLATION OF A SPRING A ball that is bobbing up and down on the end of a spring has a maimum displacement of inches. Its motion (in ideal conditions) is modeled b cos 6t t > 0, where is measured in feet and t is the time in seconds. (a) Graph the function. (b) What is the period of the oscillations? (c) Determine the first time the weight passes the point of equilibrium NUMERICAL AND GRAPHICAL ANALYSIS The cross section of an irrigation canal is an isosceles trapezoid of which of the sides are 8 feet long (see figure). The objective is to find the angle that maimizes the area of the cross section. Hint: The area of a trapezoid is hb b. Low point 8 ft 8 ft 8 ft d cos 0t d sin 79t 6 High point.5 ft

83 Section.8 (a) Complete seven additional rows of the table. Applications and Models 6 Time, t Base Base Altitude Area Sales, S cos 0 8 sin 0. Time, t cos 0 8 sin 0.5 Sales, S (b) Use a graphing utilit to generate additional rows of the table. Use the table to estimate the maimum cross-sectional area. (c) Write the area A as a function of. (d) Use a graphing utilit to graph the function. Use the graph to estimate the maimum cross-sectional area. How does our estimate compare with that of part (b)? 65. NUMERICAL AND GRAPHICAL ANALYSIS A -meter-high fence is meters from the side of a grain storage bin. A grain elevator must reach from ground level outside the fence to the storage bin (see figure). The objective is to determine the shortest elevator that meets the constraints. (a) Create a scatter plot of the data. (b) Find a trigonometric model that fits the data. Graph the model with our scatter plot. How well does the model fit the data? (c) What is the period of the model? Do ou think it is reasonable given the contet? Eplain our reasoning. (d) Interpret the meaning of the model s amplitude in the contet of the problem. 67. DATA ANALYSIS The number of hours H of dalight in Denver, Colorado on the 5th of each month are: 9.67, 0.7,.9,.5, 5.7, 6.97, 7.7, 8.77, 9.8, 0.8, 0.00, 9.8. The month is represented b t, with t corresponding to Januar. A model for the data is given b H t..77 sin t (a) Use a graphing utilit to graph the data points and the model in the same viewing window. (b) What is the period of the model? Is it what ou epected? Eplain. (c) What is the amplitude of the model? What does it represent in the contet of the problem? Eplain. L m L m (a) Complete four rows of the table. EXPLORATION L L L L 0. sin 0. cos sin 0. cos 0.. (b) Use a graphing utilit to generate additional rows of the table. Use the table to estimate the minimum length of the elevator. (c) Write the length L L as a function of. (d) Use a graphing utilit to graph the function. Use the graph to estimate the minimum length. How does our estimate compare with that of part (b)? 66. DATA ANALYSIS The table shows the average sales S (in millions of dollars) of an outerwear manufacturer for each month t, where t represents Januar. 68. CAPSTONE While walking across flat land, ou notice a wind turbine tower of height h feet directl in front of ou. The angle of elevation to the top of the tower is A degrees. After ou walk d feet closer to the tower, the angle of elevation increases to B degrees. (a) Draw a diagram to represent the situation. (b) Write an epression for the height h of the tower in terms of the angles A and B and the distance d. TRUE OR FALSE? In Eercises 69 and 70, determine whether the statement is true or false. Justif our answer. 69. The Leaning Tower of Pisa is not vertical, but if ou know the angle of elevation to the top of the tower when ou stand d feet awa from it, ou can find its height h using the formula h d tan. 70. N E means degrees north of east.

84 6 Chapter Trigonometr CHAPTER SUMMARY What Did You Learn? Eplanation/Eamples Review Eercises Describe angles (p. 80). = 0 = 0 8 Section. Section. Section. Convert between degrees and radians (p. 8). Use angles to model and solve real-life problems (p. 85). Identif a unit circle and describe its relationship to real numbers (p. 9). Evaluate trigonometric functions using the unit circle (p. 9). Use domain and period to evaluate sine and cosine functions (p. 95). Use a calculator to evaluate trigonometric functions (p. 96). Evaluate trigonometric functions of acute angles (p. 99). Use fundamental trigonometric identities (p. 0). Use a calculator to evaluate trigonometric functions (p. 0). To convert degrees to radians, multipl degrees b To convert radians to degrees, multipl radians b Angles can be used to find the length of a circular arc and the area of a sector of a circle. (See Eamples 5 and 8.) t > 0 t corresponds to cos sin, Because sin 0.99, 8 sin csc sin rad rad. 9 opp hp, hp opp,, csc (, ) t (, 0), t cos, adj hp, sec hp adj, and sin 9 sin cot tan sin, sin cos tan opp adj cot adj opp tan , csc t < 0 (, 0) t (, ),,. So tan.. cos t , Use trigonometric functions to model and solve real-life problems (p. 0). Trigonometric functions can be used to find the height of a monument, the angle between two paths, and the length of a ramp. (See Eamples 7 9.) 55, 56

85 Chapter Summar 6 Section.8 Section.7 Section.6 Section.5 Section. What Did You Learn? Eplanation/Eamples Review Eercises Evaluate trigonometric functions of an angle (p. 0). Find reference angles (p. ). Evaluate trigonometric functions of real numbers (p. ). Sketch the graphs of sine and cosine functions using amplitude and period (p. 9). Sketch translations of the graphs of sine and cosine functions (p. ). Use sine and cosine functions to model real-life data (p. 5). Sketch the graphs of tangent (p. 0), cotangent (p. ), secant (p. ), and cosecant (p. ), functions. Sketch the graphs of damped trigonometric functions (p. 5). Evaluate and graph inverse trigonometric functions (p. ). Evaluate and graph the compositions of trigonometric functions (p. 5). Solve real-life problems involving right triangles (p. 5). Solve real-life problems involving directional bearings (p. 5). Solve real-life problems involving harmonic motion (p. 5). Let, be a point on the terminal side of. Then sin and tan 5, cos 5,. Let be an angle in standard position. Its reference angle is the acute angle formed b the terminal side of and the horizontal ais. because So, cos 7 cos. For d a sinb c and d a cosb c, the constant c creates a horizontal translation. The constant d creates a vertical translation. (See Eamples 6.) A cosine function can be used to model the depth of the water at the end of a dock at various times. (See Eample 7.) cos7 = tan For f cos and g log sin, the factors and log are called damping factors. sin 6, = sin 5 7. cos, tan cosarctan 5, sinsin A trigonometric function can be used to find the height of a smokestack on top of a building. (See Eample.) Trigonometric functions can be used to find a ship s bearing and distance from a port at a given time. (See Eample 5.) Sine or cosine functions can be used to describe the motion of an object that vibrates, oscillates, rotates, or is moved b wave motion. (See Eamples 6 and 7.) = cos = sec = cos , , 0 05, 8 0 9, 0

86 6 Chapter Trigonometr REVIEW EXERCISES See for worked-out solutions to odd-numbered eercises.. In Eercises 8, (a) sketch the angle in standard position, (b) determine the quadrant in which the angle lies, and (c) determine one positive and one negative coterminal angle In Eercises 9, convert the angle measure from degrees to radians. Round our answer to three decimal places º In Eercises 6, convert the angle measure from radians to degrees. Round our answer to three decimal places In Eercises 7 0, convert each angle measure to degrees, minutes, and seconds without using a calculator ARC LENGTH Find the length of the arc on a circle with a radius of 0 inches intercepted b a central angle of 8.. PHONOGRAPH Phonograph records are vinl discs that rotate on a turntable. A tpical record album is inches in diameter and plas at revolutions per minute. (a) What is the angular speed of a record album? (b) What is the linear speed of the outer edge of a record album?. CIRCULAR SECTOR Find the area of the sector of a circle with a radius of 8 inches and central angle 0.. CIRCULAR SECTOR Find the area of the sector of a circle with a radius of 6.5 millimeters and central angle 56.. In Eercises 5 8, find the point, on the unit circle that corresponds to the real number t. 5. t 6. t 7 7. t t In Eercises 9, evaluate (if possible) the si trigonometric functions of the real number. 9. t t. t. t In Eercises 6, evaluate the trigonometric function using its period as an aid.. sin. cos 5. sin76 6. cos In Eercises 7 0, use a calculator to evaluate the trigonometric function. Round our answer to four decimal places. 7. tan 8. csc sec5 0. sin9. In Eercises and, find the eact values of the si trigonometric functions of the angle shown in the figure... 8 In Eercises 6, use the given function value and trigonometric identities (including the cofunction identities) to find the indicated trigonometric functions.. sin (a) csc (b) cos (c) sec (d) tan. tan (a) cot (b) sec (c) cos (d) csc 5. csc (a) sin (b) cos (c) sec (d) tan 6. csc (a) sin (b) cot (c) tan (d) sec In Eercises 7 5, use a calculator to evaluate the trigonometric function. Round our answer to four decimal places. 7. tan 8. csc 9. sin. 50. sec cot 5 5. csc 5 5. tan 5 5. cos RAILROAD GRADE A train travels.5 kilometers on a straight track with a grade of 0 (see figure on the net page). What is the vertical rise of the train in that distance?

87 Review Eercises 65 FIGURE FOR GUY WIRE A gu wire runs from the ground to the top of a 5-foot telephone pole. The angle formed between the wire and the ground is 5. How far from the base of the pole is the wire attached to the ground?. In Eercises 57 6, the point is on the terminal side of an angle in standard position. Determine the eact values of the si trigonometric functions of the angle. 57., 6 58., , 5, , , 0. 6.,, > 0 6.,, > 0 In Eercises 65 70, find the values of the remaining five trigonometric functions of. Function Value 65. sec 66. csc 67. sin 68. tan 69. cos 70. sin Constraint tan < 0 cos < 0 cos < 0 cos < 0 sin > 0 cos > 0 In Eercises 7 7, find the reference angle and in standard position km 65 7 and sketch In Eercises 75 80, evaluate the sine, cosine, and tangent of the angle without using a calculator In Eercises 8 8, use a calculator to evaluate the trigonometric function. Round our answer to four decimal places. 8. sin 8. cot.8 8. sin5 8. tan57.5 In Eercises 85 9, sketch the graph of the function. Include two full periods. 0 Not drawn to scale 85. sin cos 87. f 5 sin5 88. f 8 cos sin 90. cos 9. gt 5 sint 9. gt cost 9. SOUND WAVES Sound waves can be modeled b sine functions of the form a sin b, where is measured in seconds. (a) Write an equation of a sound wave whose amplitude is and whose period is second. 6 (b) What is the frequenc of the sound wave described in part (a)? 9. DATA ANALYSIS: METEOROLOGY The times S of sunset (Greenwich Mean Time) at 0 north latitude on the 5th of each month are: (6:59), (7:5), (8:06), (8:8), 5(9:08), 6(9:0), 7(9:8), 8(8:57), 9(8:09), 0(7:), (6:), (6:6). The month is represented b t, with t corresponding to Januar. A model (in which minutes have been converted to the decimal parts of an hour) for the data is St sint6.60. (a) Use a graphing utilit to graph the data points and the model in the same viewing window. (b) What is the period of the model? Is it what ou epected? Eplain. (c) What is the amplitude of the model? What does it represent in the model? Eplain..6 In Eercises 95 0, sketch a graph of the function. Include two full periods. 95. f tan 96. f t tan t 97. f cot 98. gt cot t 99. f sec 00. ht sec t 0. f 0. f t csc t csc In Eercises 0 and 0, use a graphing utilit to graph the function and the damping factor of the function in the same viewing window. Describe the behavior of the function as increases without bound. 0. f cos 0. g cos.7 In Eercises 05 0, evaluate the epression. If necessar, round our answer to two decimal places. 05. arcsin 06. arcsin 07. arcsin arcsin sin sin 0.89

88 66 Chapter Trigonometr In Eercises, evaluate the epression without using a calculator.. arccos. arccos. cos. cos In Eercises 5 8, use a calculator to evaluate the epression. Round our answer to two decimal places. 5. arccos arccos tan.5 8. tan 8. In Eercises 9, use a graphing utilit to graph the function. 9. f arcsin 0. f arccos. f arctan. f arcsin In Eercises 8, find the eact value of the epression.. cosarctan. tanarccos 5 5. sectan 6. secsin 5 7. cotarctan cotarcsin In Eercises 9 and 0, write an algebraic epression that is equivalent to the epression. 9. tanarccos 0. secarcsin In Eercises, evaluate each epression without using a calculator.. arccot. arcsec. arcsec. arccsc In Eercises 5 8, use a calculator to approimate the value of the epression. Round our result to two decimal places. 5. arccot arcsec arcsec 5 8. arccsc ANGLE OF ELEVATION The height of a radio transmission tower is 70 meters, and it casts a shadow of length 0 meters. Draw a diagram and find the angle of elevation of the sun. 0. HEIGHT Your football has landed at the edge of the roof of our school building. When ou are 5 feet from the base of the building, the angle of elevation to our football is. How high off the ground is our football?. DISTANCE From cit A to cit B, a plane flies 650 miles at a bearing of 8. From cit B to cit C, the plane flies 80 miles at a bearing of 5. Find the distance from cit A to cit C and the bearing from cit A to cit C.. WAVE MOTION Your fishing bobber oscillates in simple harmonic motion from the waves in the lake where ou fish. Your bobber moves a total of.5 inches from its high point to its low point and returns to its high point ever seconds. Write an equation modeling the motion of our bobber if it is at its high point at time t 0. EXPLORATION TRUE OR FALSE? In Eercises and, determine whether the statement is true or false. Justif our answer.. sin is not a function because sin 0 sin 50.. Because tan, arctan. 5. WRITING Describe the behavior of f sec at the zeros of g cos. Eplain our reasoning. 6. CONJECTURE (a) Use a graphing utilit to complete the table. tan cot (b) Make a conjecture about the relationship between and cot. 7. WRITING When graphing the sine and cosine functions, determining the amplitude is part of the analsis. Eplain wh this is not true for the other four trigonometric functions. 8. OSCILLATION OF A SPRING A weight is suspended from a ceiling b a steel spring. The weight is lifted (positive direction) from the equilibrium position and released. The resulting motion of the weight is modeled b Ae kt cos bt 5 et0 cos 6t, where is the distance in feet from equilibrium and t is the time in seconds. The graph of the function is shown in the figure. For each of the following, describe the change in the sstem without graphing the resulting function. (a) A is changed from 5 to. (b) k is changed from 0 to. (c) b is changed from 6 to 9. tan t

89 Chapter Test 67 CHAPTER TEST See for worked-out solutions to odd-numbered eercises. Take this test as ou would take a test in class. When ou are finished, check our work against the answers given in the back of the book. (, 6) FIGURE FOR. Consider an angle that measures radians. (a) Sketch the angle in standard position. (b) Determine two coterminal angles (one positive and one negative). (c) Convert the angle to degree measure.. A truck is moving at a rate of 05 kilometers per hour, and the diameter of its wheels is meter. Find the angular speed of the wheels in radians per minute.. A water sprinkler spras water on a lawn over a distance of 5 feet and rotates through an angle of 0. Find the area of the lawn watered b the sprinkler.. Find the eact values of the si trigonometric functions of the angle shown in the figure. 5. Given that tan find the other five trigonometric functions of. 6. Determine the reference angle for the angle and sketch and in standard position. 7. Determine the quadrant in which lies if sec < 0 and tan > Find two eact values of in degrees 0 < 60 if cos (Do not use a calculator.) 9. Use a calculator to approimate two values of in radians 0 < if csc Round the results to two decimal places..00., In Eercises 0 and, find the remaining five trigonometric functions of satisfing the conditions. 0. cos tan < 0. sec sin > 0 5, 9 0, In Eercises and, sketch the graph of the function. (Include two full periods.). g sin. f tan f FIGURE FOR 6 In Eercises and 5, use a graphing utilit to graph the function. If the function is periodic, find its period.. sin cos 5. 6e 0.t cos0.5t, 0 t 6. Find a, b, and c for the function f a sinb c such that the graph of f matches the figure. 7. Find the eact value of cotarcsin 8 without the aid of a calculator. 8. Graph the function f arcsin. 9. A plane is 90 miles south and 0 miles east of London Heathrow Airport. What bearing should be taken to fl directl to the airport? 0. Write the equation for the simple harmonic motion of a ball on a spring that starts at its lowest point of 6 inches below equilibrium, bounces to its maimum height of 6 inches above equilibrium, and returns to its lowest point in a total of seconds.

90 PROOFS IN MATHEMATICS The Pthagorean Theorem The Pthagorean Theorem is one of the most famous theorems in mathematics. More than 00 different proofs now eist. James A. Garfield, the twentieth president of the United States, developed a proof of the Pthagorean Theorem in 876. His proof, shown below, involved the fact that a trapezoid can be formed from two congruent right triangles and an isosceles right triangle. The Pthagorean Theorem In a right triangle, the sum of the squares of the lengths of the legs is equal to the square of the length of the hpotenuse, where a and b are the legs and c is the hpotenuse. a b c a c b Proof O N c b a c M b Q a P Area of Area of Area of Area of trapezoid MNOP MNQ PQO NOQ a ba b ab ab c a ba b ab c a ba b ab c a ab b ab c a b c 68

91 PROBLEM SOLVING This collection of thought-provoking and challenging eercises further eplores and epands upon concepts learned in this chapter.. The restaurant at the top of the Space Needle in Seattle, Washington is circular and has a radius of 7.5 feet. The dining part of the restaurant revolves, making about one complete revolution ever 8 minutes. A dinner part was seated at the edge of the revolving restaurant at 6:5 P.M. and was finished at 8:57 P.M. (a) Find the angle through which the dinner part rotated. (b) Find the distance the part traveled during dinner.. A biccle s gear ratio is the number of times the freewheel turns for ever one turn of the chainwheel (see figure). The table shows the numbers of teeth in the freewheel and chainwheel for the first five gears of an 8- speed touring biccle. The chainwheel completes one rotation for each gear. Find the angle through which the freewheel turns for each gear. Give our answers in both degrees and radians. Gear number 5 Freewheel. A surveor in a helicopter is tring to determine the width of an island, as shown in the figure. 000 ft Number of teeth in freewheel d Number of teeth in chainwheel w 0 Chainwheel Not drawn to scale (a) What is the shortest distance d the helicopter would have to travel to land on the island? (b) What is the horizontal distance that the helicopter would have to travel before it would be directl over the nearer end of the island? (c) Find the width w of the island. Eplain how ou obtained our answer.. Use the figure below. A (a) Eplain wh ABC, ADE, and AFG are similar triangles. (b) What does similarit impl about the ratios BC AB, DE FG, and AD AF? (c) Does the value of sin A depend on which triangle from part (a) is used to calculate it? Would the value of sin A change if it were found using a different right triangle that was similar to the three given triangles? (d) Do our conclusions from part (c) appl to the other five trigonometric functions? Eplain. 5. Use a graphing utilit to graph h, and use the graph to decide whether h is even, odd, or neither. (a) h cos (b) h sin 6. If f is an even function and g is an odd function, use the results of Eercise 5 to make a conjecture about h, where (a) h f (b) h g. 7. The model for the height h (in feet) of a Ferris wheel car is h sin 8t B C where t is the time (in minutes). (The Ferris wheel has a radius of 50 feet.) This model ields a height of 50 feet when t 0. Alter the model so that the height of the car is foot when t 0. D E F G 69

92 8. The pressure P (in millimeters of mercur) against the walls of the blood vessels of a patient is modeled b P 00 0 cos 8 t where t is time (in seconds). (a) Use a graphing utilit to graph the model. (b) What is the period of the model? What does the period tell ou about this situation? (c) What is the amplitude of the model? What does it tell ou about this situation? (d) If one ccle of this model is equivalent to one heartbeat, what is the pulse of this patient? (e) If a phsician wants this patient s pulse rate to be 6 beats per minute or less, what should the period be? What should the coefficient of t be? 9. A popular theor that attempts to eplain the ups and downs of everda life states that each of us has three ccles, called biorhthms, which begin at birth. These three ccles can be modeled b sine waves. Phsical ( das): Emotional (8 das): Intellectual ( das): I sin t t 0, where t is the number of das since birth. Consider a person who was born on Jul 0, 988. (a) Use a graphing utilit to graph the three models in the same viewing window for 700 t 780. (b) Describe the person s biorhthms during the month of September 008. (c) Calculate the person s three energ levels on September, (a) Use a graphing utilit to graph the functions given b and (b) Use the graphs from part (a) to find the period of each function. (c) If are positive integers, is the function given b h A cos B sin periodic? Eplain our reasoning.. Two trigonometric functions f and g have periods of, and their graphs intersect at 5.5. and P sin t, E sin t 8, f cos sin g cos sin. t 0 t 0 (a) Give one smaller and one larger positive value of at which the functions have the same value. 70 (b) Determine one negative value of at which the graphs intersect. (c) Is it true that f.5 g.65? Eplain our reasoning.. The function f is periodic, with period c. So, f t c f t. Are the following equal? Eplain. (a) (b) (c) f t c f t f t c f t f t c f t. If ou stand in shallow water and look at an object below the surface of the water, the object will look farther awa from ou than it reall is. This is because when light ras pass between air and water, the water refracts, or bends, the light ras. The inde of refraction for water is.. This is the ratio of the sine of and the sine of (see figure). (a) You are standing in water that is feet deep and are looking at a rock at angle 60 (measured from a line perpendicular to the surface of the water). Find. (b) Find the distances and. (c) Find the distance d between where the rock is and where it appears to be. (d) What happens to d as ou move closer to the rock? Eplain our reasoning.. In calculus, it can be shown that the arctangent function can be approimated b the polnomial arctan 5 5 ft 7 7 d where is in radians. (a) Use a graphing utilit to graph the arctangent function and its polnomial approimation in the same viewing window. How do the graphs compare? (b) Stud the pattern in the polnomial approimation of the arctangent function and guess the net term. Then repeat part (a). How does the accurac of the approimation change when additional terms are added?

93 Analtic Trigonometr 5 5. Using Fundamental Identities 5. Verifing Trigonometric Identities 5. Solving Trigonometric Equations 5. Sum and Difference Formulas 5.5 Multiple-Angle and Product-to-Sum Formulas In Mathematics Analtic trigonometr is used to simplif trigonometric epressions and solve trigonometric equations. In Real Life Analtic trigonometr is used to model real-life phenomena. For instance, when an airplane travels faster than the speed of sound, the sound waves form a cone behind the airplane. Concepts of trigonometr can be used to describe the ape angle of the cone. (See Eercise 7, page 5.) Christopher Pasatier/Reuters/Landov IN CAREERS There are man careers that use analtic trigonometr. Several are listed below. Mechanical Engineer Eercise 89, page 96 Phsicist Eercise 90, page 0 Athletic Trainer Eercise 5, page 5 Phsical Therapist Eercise 8, page 5 7

94 7 Chapter 5 Analtic Trigonometr 5. USING FUNDAMENTAL IDENTITIES What ou should learn Recognize and write the fundamental trigonometric identities. Use the fundamental trigonometric identities to evaluate trigonometric functions, simplif trigonometric epressions, and rewrite trigonometric epressions. Wh ou should learn it Fundamental trigonometric identities can be used to simplif trigonometric epressions. For instance, in Eercise on page 79, ou can use trigonometric identities to simplif an epression for the coefficient of friction. You should learn the fundamental trigonometric identities well, because the are used frequentl in trigonometr and the will also appear later in calculus. Note that u can be an angle, a real number, or a variable. Introduction In Chapter, ou studied the basic definitions, properties, graphs, and applications of the individual trigonometric functions. In this chapter, ou will learn how to use the fundamental identities to do the following.. Evaluate trigonometric functions.. Simplif trigonometric epressions.. Develop additional trigonometric identities.. Solve trigonometric equations. Fundamental Trigonometric Identities Reciprocal Identities sin u csc u csc u sin u Quotient Identities tan u sin u cos u Pthagorean Identities sin u cos u Cofunction Identities sin u cos u tan u cot u sec u csc u Even/Odd Identities sinu sin u cscu csc u cos u sec u sec u cos u cot u cos u sin u tan u sec u cos u sin u cot u tan u csc u sec u cosu cos u secu sec u tan u cot u cot u tan u cot u csc u tanu tan u cotu cot u Pthagorean identities are sometimes used in radical form such as sin u ± cos u or tan u ±sec u where the sign depends on the choice of u.

95 Section 5. Using Fundamental Identities 7 Using the Fundamental Identities One common application of trigonometric identities is to use given values of trigonometric functions to evaluate other trigonometric functions. Eample Using Identities to Evaluate a Function Use the values sec u and tan u > 0 to find the values of all si trigonometric functions. TECHNOLOGY Using a reciprocal identit, ou have cos u sec u. Using a Pthagorean identit, ou have sin u cos u Pthagorean identit Substitute for cos u. You can use a graphing utilit to check the result of Eample. To do this, graph and sin cos sin sin in the same viewing window, as shown below. Because Eample shows the equivalence algebraicall and the two graphs appear to coincide, ou can conclude that the epressions are equivalent Simplif. Because sec u < 0 and tan u > 0, it follows that u lies in Quadrant III. Moreover, because sin u is negative when u is in Quadrant III, ou can choose the negative root and obtain sin u 5. Now, knowing the values of the sine and cosine, ou can find the values of all si trigonometric functions. sin u 5 cos u tan u sin u 5 5 cos u Now tr Eercise. csc u sin u sec u cos u cot u tan u Eample Simplifing a Trigonometric Epression Simplif sin cos sin. First factor out a common monomial factor and then use a fundamental identit. sin cos sin sin cos Factor out common monomial factor. sin cos Factor out. sin sin Pthagorean identit sin Multipl. Now tr Eercise 59.

96 7 Chapter 5 Analtic Trigonometr When factoring trigonometric epressions, it is helpful to find a special polnomial factoring form that fits the epression, as shown in Eample. Eample Factoring Trigonometric Epressions In Eample, ou need to be able to factor the difference of two squares and factor a trinomial. You can review the techniques for factoring in Appendi A.. Factor each epression. a. sec b. tan a. This epression has the form u v, which is the difference of two squares. It factors as sec sec b. This epression has the polnomial form a b c, and it factors as Now tr Eercise 6. tan sec ). tan tan tan tan. On occasion, factoring or simplifing can best be done b first rewriting the epression in terms of just one trigonometric function or in terms of sine and cosine onl. These strategies are shown in Eamples and 5, respectivel. Eample Factoring a Trigonometric Epression Factor csc cot. Use the identit csc cot to rewrite the epression in terms of the cotangent. csc cot cot cot Pthagorean identit cot cot Combine like terms. cot cot Factor. Now tr Eercise 65. Eample 5 Simplifing a Trigonometric Epression Remember that when adding rational epressions, ou must first find the least common denominator (LCD). In Eample 5, the LCD is sin t. Simplif sin t cot t cos t. Begin b rewriting cot t in terms of sine and cosine. sin t cot t cos t sin t cos t sin t cos t sin t cos t sin t sin t csc t Now tr Eercise 7. Quotient identit Add fractions. Pthagorean identit Reciprocal identit

97 Section 5. Using Fundamental Identities 75 Eample 6 Adding Trigonometric Epressions Perform the addition and simplif. sin cos sin cos cos sin cos sin sin cos cos sin cos sin sin cos sin cos cos sin sin csc Now tr Eercise 75. cos cos Multipl. Pthagorean identit: sin cos Divide out common factor. Reciprocal identit The net two eamples involve techniques for rewriting epressions in forms that are used in calculus. Eample 7 Rewriting a Trigonometric Epression Rewrite sin so that it is not in fractional form. From the Pthagorean identit cos sin sin sin, ou can see that multipling both the numerator and the denominator b sin will produce a monomial denominator. sin sin sin sin sin sin sin cos sin cos cos sin cos cos cos sec tan sec Now tr Eercise 8. Multipl numerator and denominator b sin. Multipl. Pthagorean identit Write as separate fractions. Product of fractions Reciprocal and quotient identities

98 76 Chapter 5 Analtic Trigonometr Eample 8 Trigonometric Substitution Use the substitution tan, 0 < as a trigonometric function of. <, to write Begin b letting tan. Then, ou can obtain tan tan tan sec sec. Now tr Eercise 9. Substitute tan for. Rule of eponents Factor. Pthagorean identit sec > 0 for 0 < < + = arctan Angle whose tangent is. FIGURE 5. Figure 5. shows the right triangle illustration of the trigonometric substitution tan in Eample 8. You can use this triangle to check the solution of Eample 8. For 0 < <, ou have opp, adj, and hp. With these epressions, ou can write the following. sec sec sec hp adj So, the solution checks. Recall that for positive real numbers u and v, ln u ln v lnuv. You can review the properties of logarithms in Section.. Eample 9 Rewriting a Logarithmic Epression Rewrite ln csc ln tan ln csc ln tan ln csc tan as a single logarithm and simplif the result. ln sin sin cos ln cos ln sec Now tr Eercise. Product Propert of Logarithms Reciprocal and quotient identities Simplif. Reciprocal identit

99 Section 5. Using Fundamental Identities EXERCISES VOCABULARY: Fill in the blank to complete the trigonometric identit. sin u.. cos u.. tan u 5. csc u u cosu 0. tanu SKILLS AND APPLICATIONS csc u cos u See for worked-out solutions to odd-numbered eercises. tan u sec u In Eercises, use the given values to evaluate (if possible) all si trigonometric functions.. sin cos,. tan,,. sec. csc 5. tan 8 5, sin tan sec cot, sin sec csc 5, 5 8. cos 5, cos 5 9. sin tan, 0. sec, sin > 0. tan. csc. sin. tan sin cos cot is undefined, < 0 < 0 sin > 0 5 7,, 5,, cos 7 0 In Eercises 5 0, match the trigonometric epression with one of the following. (a) sec (b) (c) cot (d) (e) tan (f) sin 5. sec cos 6. tan csc 7. cot csc 8. cos csc 9. sin sin 0. cos cos In Eercises 6, match the trigonometric epression with one of the following. (a) csc (b) tan (c) sin (d) sin tan (e) sec (f) sec tan. sin sec. cos sec. sec tan. cot sec sec cos sin cos In Eercises 7 58, use the fundamental identities to simplif the epression. There is more than one correct form of each answer. 7. cot 8. cos tan 9. tan cos 0. sin cot. sin csc sin. sec sin. cot csc. csc sec 5. sin 6. csc tan 7. tan sin 8. sec tan 9. tan sec sin 50. tan sec cot sec cos 5. cos sin 5. cos t tan t 55. sin tan cos 56. csc tan sec 57. cot u sin u tan u cos u 58. sin sec cot sec cos csc csc

100 78 Chapter 5 Analtic Trigonometr In Eercises 59 70, factor the epression and use the fundamental identities to simplif. There is more than one correct form of each answer. 59. tan tan sin 60. sin csc sin 6. sin sec sin 6. cos cos tan 6. sec cos 6. sec cos 65. tan tan 66. cos cos 67. sin cos 68. sec tan csc csc csc sec sec sec In Eercises 7 7, perform the multiplication and use the fundamental identities to simplif. There is more than one correct form of each answer. 7. sin cos 7. cot csc cot csc 7. csc csc 7. sin sin In Eercises 75 80, perform the addition or subtraction and use the fundamental identities to simplif. There is more than one correct form of each answer. 75. cos cos cos sin sin cos tan cos sin 80. In Eercises 8 8, rewrite the epression so that it is not in fractional form. There is more than one correct form of each answer. 8. sin cos sec tan 8. NUMERICAL AND GRAPHICAL ANALYSIS In Eercises 85 88, use a graphing utilit to complete the table and graph the functions. Make a conjecture about and sec sec tan sec sec tan tan sec tan 5 tan sec tan csc In Eercises 89 9, use a graphing utilit to determine which of the si trigonometric functions is equal to the epression. Verif our answer algebraicall. 89. cos cot sin 90. sec csc tan In Eercises 9 0, use the trigonometric substitution to write the algebraic epression as a trigonometric function of, where 0 < < /. 9. 9, cos , cos , 9, 9,, 5, 00, 9, 9 5, sin 7 sin sec sec 5 tan 0 tan tan 5 tan 0. 0., 0, sin 0 sin In Eercises 05 08, use the trigonometric substitution to write the algebraic equation as a trigonometric equation of, where / < < /. Then find sin and cos cos, sin sec cos, sin tan sin cos sin, cos sec sec, tan tan sin cos cos sin cos cos 9, sin 6, 6 sin 6, cos 5 00, 0 cos sin In Eercises 09, use a graphing utilit to solve the equation for, where 0 <. 09. sin 0. cos. sec. csc cos sin tan cot

101 Section 5. In Eercises 8, rewrite the epression as a single logarithm and simplif the result ln cos ln sin. ln sec ln sin ln sin ln cot 6. ln tan ln csc ln cot t ln tan t ln cos t ln tan t In Eercises 9, use a calculator to demonstrate the identit for each value of. 9. csc cot (a) (b) 7 Using Fundamental Identities 79 EXPLORATION TRUE OR FALSE? In Eercises 7 and 8, determine whether the statement is true or false. Justif our answer. 7. The even and odd trigonometric identities are helpful for determining whether the value of a trigonometric function is positive or negative. 8. A cofunction identit can be used to transform a tangent function so that it can be represented b a cosecant function. In Eercises 9, fill in the blanks. (Note: The notation c ⴙ indicates that approaches c from the right and c ⴚ indicates that approaches c from the left.) 0. tan sec (a) 6 (b).. cos sin (a) 80 (b) 0.8. sin sin (a) 50 (b), sin and csc. 0. As 0, cos and sec.. As, tan and cot.. As, sin and csc. 9. As. FRICTION The forces acting on an object weighing W units on an inclined plane positioned at an angle of with the horizontal (see figure) are modeled b W cos W sin where is the coefficient of friction. Solve the equation for and simplif the result. W. RATE OF CHANGE The rate of change of the function f tan is given b the epression sec. Show that this epression can also be written as tan. 5. RATE OF CHANGE The rate of change of the function f sec cos is given b the epression sec tan sin. Show that this epression can also be written as sin tan. 6. RATE OF CHANGE The rate of change of the function f csc sin is given b the epression csc cot cos. Show that this epression can also be written as cos cot. In Eercises 8, determine whether or not the equation is an identit, and give a reason for our answer.. cos sin. cot csc sin k 5. tan, k is a constant. cos k 6. 5 sec 5 cos 7. sin csc 8. csc 9. Use the trigonometric substitution u a sin, where < < and a > 0, to simplif the epression a u. 0. Use the trigonometric substitution u a tan, where < < and a > 0, to simplif the epression a u.. Use the trigonometric substitution u a sec, where 0 < < and a > 0, to simplif the epression u a.. CAPSTONE (a) Use the definitions of sine and cosine to derive the Pthagorean identit sin cos. (b) Use the Pthagorean identit sin cos to derive the other Pthagorean identities, tan sec and cot csc. Discuss how to remember these identities and other fundamental identities.

102 80 Chapter 5 Analtic Trigonometr 5. VERIFYING TRIGONOMETRIC IDENTITIES What ou should learn Verif trigonometric identities. Wh ou should learn it You can use trigonometric identities to rewrite trigonometric equations that model real-life situations. For instance, in Eercise 70 on page 86, ou can use trigonometric identities to simplif the equation that models the length of a shadow cast b a gnomon (a device used to tell time). Introduction In this section, ou will stud techniques for verifing trigonometric identities. In the net section, ou will stud techniques for solving trigonometric equations. The ke to verifing identities and solving equations is the abilit to use the fundamental identities and the rules of algebra to rewrite trigonometric epressions. Remember that a conditional equation is an equation that is true for onl some of the values in its domain. For eample, the conditional equation sin 0 Conditional equation is true onl for n, where n is an integer. When ou find these values, ou are solving the equation. On the other hand, an equation that is true for all real values in the domain of the variable is an identit. For eample, the familiar equation sin cos Identit is true for all real numbers. So, it is an identit. Verifing Trigonometric Identities Although there are similarities, verifing that a trigonometric equation is an identit is quite different from solving an equation. There is no well-defined set of rules to follow in verifing trigonometric identities, and the process is best learned b practice. Robert W. Ginn/PhotoEdit Guidelines for Verifing Trigonometric Identities. Work with one side of the equation at a time. It is often better to work with the more complicated side first.. Look for opportunities to factor an epression, add fractions, square a binomial, or create a monomial denominator.. Look for opportunities to use the fundamental identities. Note which functions are in the final epression ou want. Sines and cosines pair up well, as do secants and tangents, and cosecants and cotangents.. If the preceding guidelines do not help, tr converting all terms to sines and cosines. 5. Alwas tr something. Even paths that lead to dead ends provide insights. Verifing trigonometric identities is a useful process if ou need to convert a trigonometric epression into a form that is more useful algebraicall. When ou verif an identit, ou cannot assume that the two sides of the equation are equal because ou are tring to verif that the are equal. As a result, when verifing identities, ou cannot use operations such as adding the same quantit to each side of the equation or cross multiplication.

103 Section 5. Verifing Trigonometric Identities 8 Eample Verifing a Trigonometric Identit Verif the identit sec sec sin. WARNING / CAUTION Remember that an identit is onl true for all real values in the domain of the variable. For instance, in Eample the identit is not true when because sec is not defined when. The left side is more complicated, so start with it. sec tan Pthagorean identit sec sec tan Simplif. sec tan cos Reciprocal identit sin Quotient identit cos cos sin Simplif. Notice how the identit is verified. You start with the left side of the equation (the more complicated side) and use the fundamental trigonometric identities to simplif it until ou obtain the right side. Now tr Eercise 5. There can be more than one wa to verif an identit. Here is another wa to verif the identit in Eample. sec sec sec sec sec cos sin Rewrite as the difference of fractions. Reciprocal identit Pthagorean identit Eample Verifing a Trigonometric Identit Verif the identit sec. sin sin Algebraic The right side is more complicated, so start with it. sin sin sin sin sin sin sin cos sec Simplif. Add fractions. Pthagorean identit Reciprocal identit Numerical Use the table feature of a graphing utilit set in radian mode to create a table that shows the values of cos and sin sin for different values of, as shown in Figure 5.. From the table, ou can see that the values appear to be identical, so sec sin sin appears to be an identit. FIGURE 5. Now tr Eercise.

104 8 Chapter 5 Analtic Trigonometr Eample Verifing a Trigonometric Identit Verif the identit tan cos tan. Algebraic B appling identities before multipling, ou obtain the following. tan cos sec sin sin cos sin cos tan Pthagorean identities Reciprocal identit Rule of eponents Quotient identit Graphical Use a graphing utilit set in radian mode to graph the left side of the identit tan cos and the right side of the identit tan in the same viewing window, as shown in Figure 5.. (Select the line stle for and the path stle for.) Because the graphs appear to coincide, tan cos tan appears to be an identit. = (tan + )(cos ) = tan FIGURE 5. Now tr Eercise 5. Eample Converting to Sines and Cosines WARNING / CAUTION Although a graphing utilit can be useful in helping to verif an identit, ou must use algebraic techniques to produce a valid proof. Verif the identit tan cot sec csc. Tr converting the left side into sines and cosines. tan cot sin cos cos sin sin cos cos sin cos sin cos sin sec csc Now tr Eercise 5. Quotient identities Add fractions. Pthagorean identit Product of fractions. Reciprocal identities As shown at the right, csc cos is considered a simplified form of cos because the epression does not contain an fractions. Recall from algebra that rationalizing the denominator using conjugates is, on occasion, a powerful simplification technique. A related form of this technique, shown below, works for simplifing trigonometric epressions as well. cos cos cos cos csc cos This technique is demonstrated in the net eample. cos cos cos sin

105 Section 5. Verifing Trigonometric Identities 8 Eample 5 Verifing a Trigonometric Identit Verif the identit sec tan cos sin. Algebraic Begin with the right side because ou can create a monomial denominator b multipling the numerator and denominator b sin. cos cos sin sin sin sin cos cos sin sin cos cos sin cos cos cos sin cos cos sin cos cos sec tan Now tr Eercise 59. Multipl numerator and denominator b sin. Multipl. Pthagorean identit Write as separate fractions. Simplif. Identities Graphical Use a graphing utilit set in the radian and dot modes to graph sec tan and cos sin in the same viewing window, as shown in Figure 5.. Because the graphs appear to coincide, sec tan cos sin appears to be an identit. 7 FIGURE 5. 5 = sec + tan 5 9 cos = sin In Eamples through 5, ou have been verifing trigonometric identities b working with one side of the equation and converting to the form given on the other side. On occasion, it is practical to work with each side separatel, to obtain one common form equivalent to both sides. This is illustrated in Eample 6. Eample 6 Working with Each Side Separatel Verif the identit cot csc sin sin. Algebraic Working with the left side, ou have cot csc Pthagorean identit csc Factor. csc csc Simplif. Now, simplifing the right side, ou have sin sin Write as separate fractions. sin sin sin csc Reciprocal identit The identit is verified because both sides are equal to csc csc csc csc.. Now tr Eercise 9.. Numerical Use the table feature of a graphing utilit set in radian mode to create a table that shows the values of cot csc and sin sin for different values of, as shown in Figure 5.5. From the table ou can see that the values appear to be identical, so cot csc sin sin appears to be an identit. FIGURE 5.5

106 8 Chapter 5 Analtic Trigonometr In Eample 7, powers of trigonometric functions are rewritten as more complicated sums of products of trigonometric functions. This is a common procedure used in calculus. Eample 7 Three Eamples from Calculus Verif each identit. a. tan tan sec tan b. sin cos cos cos 6 sin c. csc cot csc cot cot tan tan tan tan sec tan sec tan sin cos sin cos sin cos cos sin cos cos 6 sin csc cot csc csc cot csc cot cot csc cot cot Now tr Eercise 6. a. Write as separate factors. Pthagorean identit Multipl. b. Write as separate factors. Pthagorean identit Multipl. c. Write as separate factors. Pthagorean identit Multipl. CLASSROOM DISCUSSION Error Analsis You are tutoring a student in trigonometr. One of the homework problems our student encounters asks whether the following statement is an identit. tan sin? 5 6 tan Your student does not attempt to verif the equivalence algebraicall, but mistakenl uses onl a graphical approach. Using range settings of Xmin Xma Xscl / Ymin 0 Yma 0 Yscl our student graphs both sides of the epression on a graphing utilit and concludes that the statement is an identit. What is wrong with our student s reasoning? Eplain. Discuss the limitations of verifing identities graphicall.

107 Section 5. Verifing Trigonometric Identities VOCABULARY EXERCISES In Eercises and, fill in the blanks.. An equation that is true for all real values in its domain is called an. See for worked-out solutions to odd-numbered eercises.. An equation that is true for onl some values in its domain is called a. In Eercises 8, fill in the blank to complete the trigonometric identit... cot u 5. sin u cscu 8. cos u sin u cos u secu SKILLS AND APPLICATIONS In Eercises 9 50, verif the identit. 9. tan t cot t 0. sec cos. cot sec. cos sin tan sec. sin sin cos. cos sin cos 5. cos sin sin 6. sin sin cos cos 7. tan cot t sin tan 8. csc t cos tcsc t sec 9. cot t 0. tan sec csc t sin t sin t tan tan. sin cos sin 5 cos cos sin. sec 6 sec tan sec sec tan sec 5 tan. cot sec csc sin. sec cos sec csc sin cos cot sec cos sin tan tan tan cot cot sin csc sin csc sin cos cos sin cos cot csc sin sec. cos csc cot cos. cos cos sin cos tan sin cos. tan. tan tan cot 5. sec 6. cos 7. sin sin cos tan tan cot cot tan tan cot cot tan cot tan cot cos cos sin sin tan cot sin sin cos cos 0 sin sin sin cos cos cos cos sin cos cos sec cot sin t csc t tan t sec cot tansin cossin 9. tan sin tan cos cos tan sin csc sec cot

108 86 Chapter 5 Analtic Trigonometr ERROR ANALYSIS In Eercises 5 and 5, describe the error(s). 5. tan cot tan cot cot tan tan cot cot tan cot tan 5. sec sec sin tan sin tan sec sin cos sec sin sec csc sin In Eercises 5 60, (a) use a graphing utilit to graph each side of the equation to determine whether the equation is an identit, (b) use the table feature of a graphing utilit to determine whether the equation is an identit, and (c) confirm the results of parts (a) and (b) algebraicall. 5. cot cos cot sin cos cot csc 5. csc csc sin sin 55. cos cos sin cos 56. tan tan sec tan 57. csc csc cot 58. sin sin cos cos5 cot csc cos sin sin cos csc cot In Eercises 6 6, verif the identit tan5 tan sec tan sec tan tan tan sec cos sin sin sin cos sin cos cos cos In Eercises 65 68, use the cofunction identities to evaluate the epression without using a calculator. 65. sin 5 sin cos 55 cos cos 0 cos 5 cos 8 cos tan 6 cot 6 sec 7 csc RATE OF CHANGE The rate of change of the function f sin csc with respect to change in the variable is given b the epression cos csc cot. Show that the epression for the rate of change can also be cos cot. 70. SHADOW LENGTH The length s of a shadow cast b a vertical gnomon (a device used to tell time) of height h when the angle of the sun above the horizon is (see figure) can be modeled b the equation s h sin 90. sin h ft s (a) Verif that the equation for s is equal to h cot. (b) Use a graphing utilit to complete the table. Let h 5 feet s (c) Use our table from part (b) to determine the angles of the sun that result in the maimum and minimum lengths of the shadow. (d) Based on our results from part (c), what time of da do ou think it is when the angle of the sun above the horizon is 90? EXPLORATION TRUE OR FALSE? In Eercises 7 and 7, determine whether the statement is true or false. Justif our answer. 7. There can be more than one wa to verif a trigonometric identit. 7. The equation sin cos tan is an identit because sin 0 cos 0 and tan 0. THINK ABOUT IT In Eercises 7 77, eplain wh the equation is not an identit and find one value of the variable for which the equation is not true. 7. sin cos 75. cos sin 77. tan sec 7. tan sec 76. csc cot 78. CAPSTONE Write a short paper in our own words eplaining to a classmate the difference between a trigonometric identit and a conditional equation. Include suggestions on how to verif a trigonometric identit.

109 Section 5. Solving Trigonometric Equations SOLVING TRIGONOMETRIC EQUATIONS Tom Stillo/Inde Stock Imager/Photo Librar What ou should learn Use standard algebraic techniques to solve trigonometric equations. Solve trigonometric equations of quadratic tpe. Solve trigonometric equations involving multiple angles. Use inverse trigonometric functions to solve trigonometric equations. Wh ou should learn it You can use trigonometric equations to solve a variet of real-life problems. For instance, in Eercise 9 on page 96, ou can solve a trigonometric equation to help answer questions about monthl sales of skiing equipment. Introduction To solve a trigonometric equation, use standard algebraic techniques such as collecting like terms and factoring. Your preliminar goal in solving a trigonometric equation is to isolate the trigonometric function in the equation. For eample, to solve the equation sin, divide each side b to obtain sin. To solve for, note in Figure 5.6 that the equation sin has solutions 6 and 56 in the interval 0,. Moreover, because sin has a period of, there are infinitel man other solutions, which can be written as n 6 and where n is an integer, as shown in Figure 5.6. = 6 FIGURE 5.6 = 5 n 6 = 6 = 5 = General solution = sin = + 6 = Another wa to show that the equation sin has infinitel man solutions is indicated in Figure 5.7. An angles that are coterminal with 6 or 56 will also be solutions of the equation. ( ) sin ( + n) = sin 5 + n = FIGURE 5.7 When solving trigonometric equations, ou should write our answer(s) using eact values rather than decimal approimations.

110 88 Chapter 5 Analtic Trigonometr Eample Collecting Like Terms Solve sin sin. Begin b rewriting the equation so that sin is isolated on one side of the equation. Write original equation. Add sin to each side. Subtract from each side. Combine like terms. Divide each side b. Because sin has a period of, first find all solutions in the interval 0,. These solutions are 5 and 7. Finall, add multiples of to each of these solutions to get the general form 5 n where n is an integer. sin sin sin sin 0 sin sin sin sin and Now tr Eercise. 7 n General solution Eample Etracting Square Roots Solve tan 0. WARNING / CAUTION When ou etract square roots, make sure ou account for both the positive and negative solutions. Begin b rewriting the equation so that tan is isolated on one side of the equation. tan 0 Write original equation. tan Add to each side. tan Divide each side b. Etract square roots. Because tan has a period of, first find all solutions in the interval 0,. These solutions are 6 and 56. Finall, add multiples of to each of these solutions to get the general form n 6 tan ± where n is an integer. and ± Now tr Eercise 5. 5 n 6 General solution

111 Section 5. Solving Trigonometric Equations 89 The equations in Eamples and involved onl one trigonometric function. When two or more functions occur in the same equation, collect all terms on one side and tr to separate the functions b factoring or b using appropriate identities. This ma produce factors that ield no solutions, as illustrated in Eample. Eample Factoring FIGURE 5.8 = cot cos cot Solve cot cos cot. Begin b rewriting the equation so that all terms are collected on one side of the equation. cot cos cot cot cos cot 0 cot cos 0 Write original equation. Subtract cot from each side. Factor. B setting each of these factors equal to zero, ou obtain cot 0 and cos 0 cos cos ±. The equation cot 0 has the solution [in the interval 0, ]. No solution is obtained for cos ± because ± are outside the range of the cosine function. Because cot has a period of, the general form of the solution is obtained b adding multiples of to, to get n General solution where n is an integer. You can confirm this graphicall b sketching the graph of cot cos cot, as shown in Figure 5.8. From the graph ou can see that the -intercepts occur at,,,, and so on. These -intercepts correspond to the solutions of cot cos cot 0. Now tr Eercise 9. You can review the techniques for solving quadratic equations in Appendi A.5. Equations of Quadratic Tpe Man trigonometric equations are of quadratic tpe a b c 0. Here are a couple of eamples. Quadratic in sin sin sin 0 sin sin 0 Quadratic in sec sec sec 0 sec sec 0 To solve equations of this tpe, factor the quadratic or, if this is not possible, use the Quadratic Formula.

112 90 Chapter 5 Analtic Trigonometr Eample Factoring an Equation of Quadratic Tpe Find all solutions of sin sin 0 in the interval 0,. Algebraic Begin b treating the equation as a quadratic in sin and factoring. sin sin 0 sin sin 0 Write original equation. Factor. Setting each factor equal to zero, ou obtain the following solutions in the interval 0,. sin 0 and sin 0 Graphical Use a graphing utilit set in radian mode to graph sin sin for 0 <, as shown in Figure 5.9. Use the zero or root feature or the zoom and trace features to approimate the -intercepts to be.57, , and These values are the approimate solutions of sin sin 0 in the interval 0,. sin sin = sin sin 7 6, 6 0 Now tr Eercise. FIGURE 5.9 Eample 5 Rewriting with a Single Trigonometric Function Solve sin cos 0. This equation contains both sine and cosine functions. You can rewrite the equation so that it has onl cosine functions b using the identit sin cos. sin cos 0 Write original equation. cos cos 0 Pthagorean identit cos cos 0 Multipl each side b. cos cos 0 Factor. Set each factor equal to zero to find the solutions in the interval 0,. cos 0 cos 0 Because cos has a period of, the general form of the solution is obtained b adding multiples of to get n, where n is an integer. n, Now tr Eercise 5. cos cos 5 n, 5 0 General solution

113 Section 5. Solving Trigonometric Equations 9 Sometimes ou must square each side of an equation to obtain a quadratic, as demonstrated in the net eample. Because this procedure can introduce etraneous solutions, ou should check an solutions in the original equation to see whether the are valid or etraneous. Eample 6 Squaring and Converting to Quadratic Tpe Find all solutions of cos sin in the interval 0,. You square each side of the equation in Eample 6 because the squares of the sine and cosine functions are related b a Pthagorean identit. The same is true for the squares of the secant and tangent functions and for the squares of the cosecant and cotangent functions. It is not clear how to rewrite this equation in terms of a single trigonometric function. Notice what happens when ou square each side of the equation. cos sin cos cos sin cos cos cos cos cos cos 0 cos cos 0 cos cos 0 Write original equation. Square each side. Pthagorean identit Rewrite equation. Combine like terms. Factor. Setting each factor equal to zero produces cos 0 and cos 0 cos 0 cos., Because ou squared the original equation, check for etraneous solutions. Check / cos? sin 0 Check / cos? sin 0 Substitute for. checks. Substitute for. does not check. Check cos? sin 0 Substitute for. checks. Of the three possible solutions, is etraneous. So, in the interval 0,, the onl two solutions are and. Now tr Eercise 7.

114 9 Chapter 5 Analtic Trigonometr Functions Involving Multiple Angles The net two eamples involve trigonometric functions of multiple angles of the forms sin ku and cos ku. To solve equations of these forms, first solve the equation for ku, then divide our result b k. Eample 7 Functions of Multiple Angles Solve cos t 0. cos t 0 cos t Write original equation. Add to each side. cos t Divide each side b. In the interval 0,, ou know that t and t 5 are the onl solutions, so, in general, ou have t n and t 5 n. Dividing these results b, ou obtain the general solution n and t 5 n t General solution 9 9 where n is an integer. Now tr Eercise 9. Eample 8 Functions of Multiple Angles Solve tan 0. tan 0 tan tan Write original equation. Subtract from each side. Divide each side b. In the interval 0,, ou know that is the onl solution, so, in general, ou have n. Multipling this result b, ou obtain the general solution n where n is an integer. Now tr Eercise. General solution

115 Section 5. Solving Trigonometric Equations 9 Using Inverse Functions In the net eample, ou will see how inverse trigonometric functions can be used to solve an equation. Eample 9 Using Inverse Functions Solve sec tan. sec tan Write original equation. tan tan 0 Pthagorean identit tan tan 0 Combine like terms. tan tan 0 Factor. Setting each factor equal to zero, ou obtain two solutions in the interval,. [Recall that the range of the inverse tangent function is,.] tan 0 tan arctan and tan 0 tan Finall, because tan has a period of, ou obtain the general solution b adding multiples of arctan n and n General solution where n is an integer. You can use a calculator to approimate the value of arctan. Now tr Eercise 6. CLASSROOM DISCUSSION Equations with No s One of the following equations has solutions and the other two do not. Which two equations do not have solutions? a. sin 5 sin 6 0 b. sin sin 6 0 c. sin 5 sin 6 0 Find conditions involving the constants b and c that will guarantee that the equation sin b sin c 0 has at least one solution on some interval of length.

116 9 Chapter 5 Analtic Trigonometr 5. EXERCISES VOCABULARY: Fill in the blanks. See for worked-out solutions to odd-numbered eercises.. When solving a trigonometric equation, the preliminar goal is to the trigonometric function involved in the equation.. The equation sin 0 has the solutions n and n, which are 6 6 called solutions.. The equation tan tan 0 is a trigonometric equation that is of tpe.. A solution of an equation that does not satisf the original equation is called an solution. SKILLS AND APPLICATIONS 7 In Eercises 5 0, verif that the -values are solutions of the equation cos 0 (a) (a) (a) (a) (a) (a) sec 0 tan 0 cos 0 6 sin sin 0 csc csc 0 6 In Eercises, solve the equation.. cos 0. sin 0. csc 0. tan 0 5. sec 0 6. cot 0 7. sin sin 0 8. tan tan 0 9. cos 0 0. sin cos. sin. tan. tan tan 0. cos cos 0 In Eercises 5 8, find all solutions of the equation in the interval [0,. 5. cos cos 6. sec 0 (b) (b) (b) (b) (b) (b) tan tan 8. sin cos 9. sec sec 0. sec csc csc. sin csc 0. sec tan. cos cos sin sin 0 sec tan 0 cos sin tan 7. csc cot 8. sin cos In Eercises 9, solve the multiple-angle equation. 9. cos 0.. tan. sec. cos. In Eercises 5 8, find the -intercepts of the graph. 5. sin 6. sin cos sec 8 tan 6 sin sin 5

117 Section 5. Solving Trigonometric Equations 95 In Eercises 9 58, use a graphing utilit to approimate the solutions (to three decimal places) of the equation in the interval [0, sin cos 0 sin sin sin 0 sin cos cot cos 5. cos sin sin 5. tan 0 5. cos sec 0.5 tan 0 csc 0.5 cot 5 0 tan 7 tan sin 7 sin 0 In Eercises 59 6, use the Quadratic Formula to solve the equation in the interval [0,. Then use a graphing utilit to approimate the angle. 59. sin sin tan tan 0 6. tan tan 0 6. cos cos 0 In Eercises 6 7, use inverse functions where needed to find all solutions of the equation in the interval [0,. 6. tan tan 0 6. tan tan tan 6 tan sec tan cos 5 cos sin 7 sin cot cot 6 cot sec sec 0 7. sec sec csc csc 0 7. csc 5 csc 0 In Eercises 75 78, use a graphing utilit to approimate the solutions (to three decimal places) of the equation in the given interval tan 5 tan 0, cos cos 0, 0, cos sin 0,, sec tan 6 0,,, In Eercises 79 8, (a) use a graphing utilit to graph the function and approimate the maimum and minimum points on the graph in the interval [0,, and (b) solve the trigonometric equation and demonstrate that its solutions are the -coordinates of the maimum and minimum points of f. (Calculus is required to find the trigonometric equation.) Function 79. f sin cos 80. f cos sin 8. f sin cos 8. f sin cos 8. f sin cos 8. f sec tan Trigonometric Equation FIXED POINT In Eercises 85 and 86, find the smallest positive fied point of the function f. [ A fied point of a function f is a real number c such that fc c.] 85. f tan 86. f cos 87. GRAPHICAL REASONING Consider the function given b f cos and its graph shown in the figure. (a) What is the domain of the function? (b) Identif an smmetr and an asmptotes of the graph. (c) Describe the behavior of the function as 0. (d) How man solutions does the equation cos 0 sec tan sec 0 sin cos sin 0 sin cos cos 0 cos sin 0 cos sin cos 0 sin cos 0 have in the interval,? Find the solutions. (e) Does the equation cos 0 have a greatest solution? If so, approimate the solution. If not, eplain wh.

118 96 Chapter 5 Analtic Trigonometr 88. GRAPHICAL REASONING Consider the function given b f sin and its graph shown in the figure. (a) What is the domain of the function? (b) Identif an smmetr and an asmptotes of the graph. (c) Describe the behavior of the function as 0. (d) How man solutions does the equation sin 0 have in the interval 8, 8? Find the solutions. 89. HARMONIC MOTION A weight is oscillating on the end of a spring (see figure). The position of the weight relative to the point of equilibrium is given b cos 8t sin 8t, where is the displacement (in meters) and t is the time (in seconds). Find the times when the weight is at the point of equilibrium 0 for 0 t. 9. SALES The monthl sales S (in hundreds of units) of skiing equipment at a sports store are approimated b S cos where t is the time (in months), with t corresponding to Januar. Determine the months in which sales eceed 7500 units. 9. PROJECTILE MOTION A batted baseball leaves the bat at an angle of with the horizontal and an initial velocit of v 0 00 feet per second. The ball is caught b an outfielder 00 feet from home plate (see figure). Find if the range r of a projectile is given b r v 0 sin. 9. PROJECTILE MOTION A sharpshooter intends to hit a target at a distance of 000 ards with a gun that has a muzzle velocit of 00 feet per second (see figure). Neglecting air resistance, determine the gun s minimum angle of elevation if the range r is given b r v 0 sin. t 6 r = 00 ft Not drawn to scale Equilibrium r = 000 d Not drawn to scale 90. DAMPED HARMONIC MOTION The displacement from equilibrium of a weight oscillating on the end of a spring is given b.56e 0.t cos.9t, where is the displacement (in feet) and t is the time (in seconds). Use a graphing utilit to graph the displacement function for 0 t 0. Find the time beond which the displacement does not eceed foot from equilibrium. 9. SALES The monthl sales S (in thousands of units) of a seasonal product are approimated b S sin t 6 where t is the time (in months), with t corresponding to Januar. Determine the months in which sales eceed 00,000 units. 95. FERRIS WHEEL A Ferris wheel is built such that the height h (in feet) above ground of a seat on the wheel at time t (in minutes) can be modeled b ht 5 50 sin 6 t. The wheel makes one revolution ever seconds. The ride begins when t 0. (a) During the first seconds of the ride, when will a person on the Ferris wheel be 5 feet above ground? (b) When will a person be at the top of the Ferris wheel for the first time during the ride? If the ride lasts 60 seconds, how man times will a person be at the top of the ride, and at what times?

119 Section DATA ANALYSIS: METEOROLOGY The table shows the average dail high temperatures in Houston H (in degrees Fahrenheit) for month t, with t corresponding to Januar. (Source: National Climatic Data Center) Month, t Houston, H (b) A quadratic approimation agreeing with f at 5 is g Use a graphing utilit to graph f and g in the same viewing window. Describe the result. (c) Use the Quadratic Formula to find the zeros of g. Compare the zero in the interval 0, 6 with the result of part (a). TRUE OR FALSE? In Eercises 99 and 00, determine whether the statement is true or false. Justif our answer. 99. The equation sin t 0 has four times the number of solutions in the interval 0, as the equation sin t If ou correctl solve a trigonometric equation to the statement sin., then ou can finish solving the equation b using an inverse function. 97. GEOMETRY The area of a rectangle (see figure) inscribed in one arc of the graph of cos is given b A cos, 0 < <. 97 EXPLORATION (a) Create a scatter plot of the data. (b) Find a cosine model for the temperatures in Houston. (c) Use a graphing utilit to graph the data points and the model for the temperatures in Houston. How well does the model fit the data? (d) What is the overall average dail high temperature in Houston? (e) Use a graphing utilit to describe the months during which the average dail high temperature is above 86 F and below 86 F. Solving Trigonometric Equations (a) Use a graphing utilit to graph the area function, and approimate the area of the largest inscribed rectangle. (b) Determine the values of for which A. 98. QUADRATIC APPROXIMATION Consider the function given b f sin 0.6. (a) Approimate the zero of the function in the interval 0, THINK ABOUT IT Eplain what would happen if ou divided each side of the equation cot cos cot b cot. Is this a correct method to use when solving equations? 0. GRAPHICAL REASONING Use a graphing utilit to confirm the solutions found in Eample 6 in two different was. (a) Graph both sides of the equation and find the -coordinates of the points at which the graphs intersect. Left side: cos Right side: sin (b) Graph the equation cos sin and find the -intercepts of the graph. Do both methods produce the same -values? Which method do ou prefer? Eplain. 0. Eplain in our own words how knowledge of algebra is important when solving trigonometric equations. 0. CAPSTONE Consider the equation sin 0. Eplain the similarities and differences between finding all solutions in the interval 0,, finding all solutions in the interval 0,, and finding the general solution. PROJECT: METEOROLOGY To work an etended application analzing the normal dail high temperatures in Phoeni and in Seattle, visit this tet s website at academic.cengage.com. (Data Source: NOAA)

120 98 Chapter 5 Analtic Trigonometr 5. SUM AND DIFFERENCE FORMULAS What ou should learn Use sum and difference formulas to evaluate trigonometric functions, verif identities, and solve trigonometric equations. Wh ou should learn it You can use identities to rewrite trigonometric epressions. For instance, in Eercise 89 on page 0, ou can use an identit to rewrite a trigonometric epression in a form that helps ou analze a harmonic motion equation. Using Sum and Difference Formulas In this and the following section, ou will stud the uses of several trigonometric identities and formulas. Sum and Difference Formulas sinu v sin u cos v cos u sin v sinu v sin u cos v cos u sin v cosu v cos u cos v sin u sin v cosu v cos u cos v sin u sin v tanu v tan u tan v tan u tan v tanu v tan u tan v tan u tan v For a proof of the sum and difference formulas, see Proofs in Mathematics on page. Eamples and show how sum and difference formulas can be used to find eact values of trigonometric functions involving sums or differences of special angles. Eample Evaluating a Trigonometric Function Richard Megna/Fundamental Photographs Find the eact value of sin. To find the eact value of sin use the fact that,. Consequentl, the formula for sinu v ields sin sin sin cos cos sin 6. Tr checking this result on our calculator. You will find that sin Now tr Eercise 7.

121 Section 5. Sum and Difference Formulas 99 Eample Evaluating a Trigonometric Function Another wa to solve Eample is to use the fact that together with the formula for cosu v. 5 u Find the eact value of cos 75. Using the fact that , together with the formula for cosu v, ou obtain cos 75 cos0 5 Eample cos 0 cos 5 sin 0 sin 5 Now tr Eercise. 6. Evaluating a Trigonometric Epression 5 = Find the eact value of sinu v given FIGURE 5.0 = 5 v sin u where 0 < u < and cos v where < v <. 5,,, Because sin u 5 and u is in Quadrant I, cos u 5, as shown in Figure 5.0. Because cos v and v is in Quadrant II, sin v 5, as shown in Figure 5.. You can find sinu v as follows. sinu v sin u cos v cos u sin v FIGURE Now tr Eercise. Eample An Application of a Sum Formula u v FIGURE 5. Write cosarctan arccos as an algebraic epression. This epression fits the formula for cosu v. Angles u arctan and v arccos are shown in Figure 5.. So cosu v cosarctan cosarccos sinarctan sinarccos. Now tr Eercise 57.

122 00 Chapter 5 Analtic Trigonometr HISTORICAL NOTE The Granger Collection, New York Hipparchus, considered the most eminent of Greek astronomers, was born about 90 B.C. in Nicaea. He was credited with the invention of trigonometr. He also derived the sum and difference formulas for sina ± B and cosa ± B. Eample 5 shows how to use a difference formula to prove the cofunction identit cos sin. Eample 5 Prove the cofunction identit Proving a Cofunction Identit Using the formula for cosu v, ou have cos cos cos sin sin 0cos sin sin. Now tr Eercise 6. cos sin. Sum and difference formulas can be used to rewrite epressions such as sin n and cos n where n is an integer, as epressions involving onl sin or cos. The resulting formulas are called reduction formulas. Eample 6 Deriving Reduction Formulas Simplif each epression. a. cos b. tan a. Using the formula for cosu v, ou have cos cos cos sin cos 0 sin sin. b. Using the formula for tanu v, ou have tan tan tan tan tan tan 0 tan 0 tan. Now tr Eercise 7. sin

123 Section 5. Sum and Difference Formulas 0 Eample 7 Solving a Trigonometric Equation Algebraic Using sum and difference formulas, rewrite the equation as Find all solutions of sin in the interval 0,. sin sin cos cos sin sin cos cos sin sin cos sin sin Graphical Sketch the graph of sin for 0 <. as shown in Figure 5.. From the graph ou can see that the -intercepts are 5 and 7. So, the solutions in the interval 0, are 5 and sin 7. sin. So, the onl solutions in the interval 0, are 5 and 7. Now tr Eercise 79. FIGURE 5. ( ( ( = sin + + sin + ( The net eample was taken from calculus. It is used to derive the derivative of the sine function. Eample 8 An Application from Calculus Verif that sin h sin h Using the formula for sinu v, ou have sin h sin h sin h cos h sin cos h h. Now tr Eercise 05. sin h cos h sin sin cos h cos sin h sin h cos sin h sin cos h h cos h h where h 0.

124 0 Chapter 5 Analtic Trigonometr 5. EXERCISES See for worked-out solutions to odd-numbered eercises. VOCABULARY: Fill in the blank.. sinu v. cosu v. tanu v. sinu v 5. cosu v 6. tanu v SKILLS AND APPLICATIONS In Eercises 7, find the eact value of each epression. In Eercises 8, find the eact values of the sine, cosine, and tangent of the angle In Eercises 9 6, write the epression as the sine, cosine, or tangent of an angle. 9. sin cos. cos sin. 7. (a) cos (b) cos cos 5 8. (a) (b) sin 5 sin sin (a) sin (b) sin 7 sin (a) cos0 5 (b) cos 0 cos 5. (a) sin5 0 (b) sin 5 cos 0. (a) sin5 60 (b) sin 5 sin cos 7 cos sin 5 7 sin 5. sin 60 cos 5 cos 60 sin 5. cos 0 cos 0 sin 0 sin 0 tan 5 tan 0. tan 5 tan 0 tan 0 tan 60. tan 0 tan tan tan tan tan 6. cos cos sin sin In Eercises 7, find the eact value of the epression. 7. sin cos cos sin 8. cos cos sin sin sin 0 cos 60 cos 0 sin cos 0 cos 0 sin 0 sin 0 tan56 tan6. tan56 tan6 tan 5 tan 0. tan 5 tan 0 In Eercises 50, find the eact value of the trigonometric function given that and cos v 5. (Both u and v are in Quadrant II.). sinu v. cosu v 5. cosu v 6. sinv u 7. tanu v 8. cscu v 9. secv u 50. cotu v In Eercises 5 56, find the eact value of the trigonometric function given that and cos v 5 5. (Both u and v are in Quadrant III.) 5. cosu v 5. sinu v 5. tanu v 5. cotv u 55. cscu v 56. secv u In Eercises 57 60, write the trigonometric epression as an algebraic epression. 57. sinarcsin arccos 58. sinarctan arccos 59. cosarccos arcsin 60. cosarccos arctan

125 Section 5. Sum and Difference Formulas 0 In Eercises 6 70, prove the identit sin sin cos cos 6. In Eercises 7 7, simplif the epression algebraicall and use a graphing utilit to confirm our answer graphicall. 7. cos sin 7. In Eercises 75 8, find all solutions of the equation in the interval 0,. 75. sin sin sin sin cos cos cos cos In Eercises 85 88, use a graphing utilit to approimate the solutions in the interval 0,. 85. sin 6 cos sin 6. cos sin 65. sin tan tan tan 67. cos cos cos sin 68. sin sin sin sin 69. sin sin sin cos 70. cos cos cos cos cos 5 cos sin sin cos cos tan sin 0 8. sin cos 0 8. cos sin 0 cos 6 sin sin cos 6 cos tan 86. tan cos sin cos cos sin HARMONIC MOTION A weight is attached to a spring suspended verticall from a ceiling. When a driving force is applied to the sstem, the weight moves verticall from its equilibrium position, and this motion is modeled b sin t cos t where is the distance from equilibrium (in feet) and t is the time (in seconds). (a) Use the identit where C arctanba, a > 0, to write the model in the form a b sinbt C. (b) Find the amplitude of the oscillations of the weight. (c) Find the frequenc of the oscillations of the weight. 90. STANDING WAVES The equation of a standing wave is obtained b adding the displacements of two waves traveling in opposite directions (see figure). Assume that each of the waves has amplitude A, period T, and wavelength. If the models for these waves are A cos t T show that A cos t T t = 0 t = t = a sin B b cos B a b sinb C T 8 T 8 and cos A cos t T

126 0 Chapter 5 Analtic Trigonometr EXPLORATION TRUE OR FALSE? In Eercises 9 9, determine whether the statement is true or false. Justif our answer. 9. sin u ± v sin u cos v ± cos u sin v 9. cos u ± v cos u cos v ± sin u sin v 9. tan tan tan 9. sin cos 0.5 h 95. cos n n cos, n is an integer 96. sin n n sin, n is an integer 97. a sin B b cos B a b sin B C, where C arctan b a and a > a sin B b cos B a b cos B C, where C arctan a b and b > sin cos 0. sin 5 cos (b) a ⴙ b cos B ⴚ C 00. sin cos 0. sin cos In Eercises 0 and 0, use the formulas given in Eercises 97 and 98 to write the trigonometric epression in the form a sin B ⴙ b cos B. 0. sin 0. 5 cos 05. Verif the following identit used in calculus. cos h cos h cos cos h sin sin h h h 06. Let 6 in the identit in Eercise 05 and define the functions f and g as follows. f h cos 6 h cos 6 h g h cos f h g h In Eercises 07 and 08, use the figure, which shows two lines whose equations are ⴝ m ⴙ b and ⴝ m ⴙ b. Assume that both lines have positive slopes. Derive a formula for the angle between the two lines. Then use our formula to find the angle between the given pair of lines. 6 = m + b In Eercises 99 0, use the formulas given in Eercises 97 and 98 to write the trigonometric epression in the following forms. a ⴙ b sin B ⴙ C 0. (c) Use a graphing utilit to graph the functions f and g. (d) Use the table and the graphs to make a conjecture about the values of the functions f and g as h 0. In Eercises 95 98, verif the identit. (a) 0. cos h sin h sin 6 h 6 h (a) What are the domains of the functions f and g? (b) Use a graphing utilit to complete the table. = m + b 07. and 08. and In Eercises 09 and 0, use a graphing utilit to graph and in the same viewing window. Use the graphs to determine whether ⴝ. Eplain our reasoning. 09. cos, cos cos 0. sin, sin sin. PROOF (a) Write a proof of the formula for sin u v. (b) Write a proof of the formula for sin u v.. CAPSTONE Give an eample to justif each statement. (a) sin u v sin u sin v (b) sin u v sin u sin v (c) cos u v cos u cos v (d) cos u v cos u cos v (e) tan u v tan u tan v (f) tan u v tan u tan v

127 Section 5.5 Multiple-Angle and Product-to-Sum Formulas MULTIPLE-ANGLE AND PRODUCT-TO-SUM FORMULAS What ou should learn Use multiple-angle formulas to rewrite and evaluate trigonometric functions. Use power-reducing formulas to rewrite and evaluate trigonometric functions. Use half-angle formulas to rewrite and evaluate trigonometric functions. Use product-to-sum and sum-toproduct formulas to rewrite and evaluate trigonometric functions. Use trigonometric formulas to rewrite real-life models. Wh ou should learn it You can use a variet of trigonometric formulas to rewrite trigonometric functions in more convenient forms. For instance, in Eercise 5 on page 5, ou can use a double-angle formula to determine at what angle an athlete must throw a javelin. Multiple-Angle Formulas In this section, ou will stud four other categories of trigonometric identities.. The first categor involves functions of multiple angles such as sin ku and cos ku.. The second categor involves squares of trigonometric functions such as sin u.. The third categor involves functions of half-angles such as sinu.. The fourth categor involves products of trigonometric functions such as sin u cos v. You should learn the double-angle formulas because the are used often in trigonometr and calculus. For proofs of these formulas, see Proofs in Mathematics on page. Double-Angle Formulas sin u sin u cos u tan u Eample tan u tan u cos u cos u sin u Solving a Multiple-Angle Equation cos u sin u Solve cos sin 0. Mark Dadswell/Gett Images Begin b rewriting the equation so that it involves functions of rather than. Then factor and solve. cos 0 and sin 0, So, the general solution is n cos sin 0 cos sin cos 0 cos sin 0 and Write original equation. Double-angle formula Factor. Set factors equal to zero. s in 0, where n is an integer. Tr verifing these solutions graphicall. Now tr Eercise 9. n