4.2 TRIGONOMETRIC FUNCTIONS: THE UNIT CIRCLE

Transcription

1 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 ug the unit circle. Use the domain and period to evaluate e and coe 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 insce, in Eercise 0 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 sdard 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.

2 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. e ecant coe secant gent cogent These si functions are normall abbreviated, csc,, sec,, 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. t csc t, 0 t sec t, 0 t, cot t, 0 0 (0, ),, ( ), (, 0) ( ) FIGURE. (, ) ( ), ( ),, (, 0) ( ) FIGURE. (0, ) (0, ) (0, ) ( ) (, 0) ( ), ( ), (, 0) ( ), (, ) ( ), In the definitions of the trigonometric functions, note that the gent and secant are not defined when 0. For insce, because t corresponds to, 0,, it follows that and sec are undefined. Similarl, the cogent and ecant are not defined when 0. For insce, 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 0,,,,,, Similarl, in Figure., the unit circle has been divided into equal arcs, corresponding to t-values of 0, and.,,,, 5,,,,,,, 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 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.. Ug 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.

3 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 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 5 b. t corresponds to the point 5 5 5,,. c. t 0 corresponds to the point,, 0.,,. csc sec cot csc 5 sec 5 cot csc 0 is undefined. 0 sec cot 0 is undefined. d. t corresponds to the point,, Now tr Eercise. csc is undefined. sec cot is undefined.

4 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,,. Now tr Eercise. csc sec cot (0, ) (, 0) (, 0) (0, ) FIGURE. π π π t =, + π, + π,... π π π π t =, + π t =, +,... π,... t = π, π,... t = 0, π,... 5π 5π t =, + π,... 7π 7π t =, + π,... t = π, π + π, π + π,... FIGURE.5 Domain and Period of Sine and Coe The domain of the e and coe 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, t and t. Because, is on the unit circle, ou know that and. So, the values of e and coe 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 t and t correspond to those of t and t. Similar results can be obtained for repeated revolutions (positive or negative) on the unit circle. This leads to the general result and t t n t t n t 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.

5 9 Chapter Trigonometr Recall from Section.5 that a function f is even if f t f t, and is odd if f t f t. Even and Odd Trigonometric Functions The coe and secant functions are even. t t sect sec t The e, ecant, gent, and cogent functions are odd. t t csct csc t t t cott cot t Eample Ug the Period to Evaluate the Sine and Coe From the definition of periodic function, it follows that the e and coe functions are periodic and have a period of. The other four trigonometric functions are also periodic, and will be discussed further in Section.. TECHNOLOGY When evaluating trigonometric functions with a calculator, remember to enclose all fractional angle measures in parentheses. For insce, if ou want to evaluate t for t /, ou should enter. 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, b. Because 7 ou have, c. For t t because the e 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 ecant, secant, and cogent functions. To evaluate these functions, ou can use the ke with their respective reciprocal functions e, coe, and gent. For insce, to evaluate csc8, use the fact that csc 8 8 and enter the following kestroke sequence in radian mode. SIN 8 ENTER Displa.59 Eample 7 Ug a Calculator Function Mode Calculator Kestrokes Displa a. Radian SIN ENTER b. cot.5 Radian TAN.5 ENTER Now tr Eercise

6 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 f t and 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 5 In Eercises 9, find the point, on the unit circle that corresponds to the real number t. 9. t 0. t. t. t. t t. In Eercises 7, evaluate (if possible) the e, coe, and gent of the real number. 7. t 8. t 9. t 0. t. t 7. t t 5 t ( 5, ( ( ( t ( 8 5, 7 7 t ( ( t. t. 5. t. 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 ug its period as an aid In Eercises 8, use the value of the trigonometric function to evaluate the indicated functions.. t. t 8 (a) t (a) t (b) csct (b) csc t 5. t 5. t (a) t (a) t (b) sect (b) sect 7. t 5 8. t 5 (a) (a) (b) t (b) t t t 5 t 5 t 7 t t

7 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.) 5. cot HARMONIC MOTION The displacement from equilibrium of an oscillating weight suspended b a spring is given b t t, 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,. 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 et t, where is the displacement (in feet) and t is the time (in seconds). (a) Complete the table. (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 increases? EXPLORATION TRUE OR FALSE? In Eercises, determine whether the statement is true or false. Justif our answer.. Because t t, it can be said that the e of a negative angle is a negative number.. a a. The real number 0 corresponds to the point 0, on the unit circle csc sec sec cot0.9 t 0 7 csc 5. Let, and, be points on the unit circle corresponding to t t and t t, respectivel. (a) Identif the smmetr of the points, and,. t (b) Make a conjecture about an relationship between t and (c) Make a conjecture about an relationship between t and. Use the unit circle to verif that the coe and secant functions are even and that the e, ecant, gent, and cogent functions are odd. t. t. 7. Verif that t t b approimating.5 and Verif that t t t t b approimating 0.5, 0.75, and. 9. THINK ABOUT IT Because ft t is an odd function and gt t is an even function, what can be said about the function ht ftgt? 70. THINK ABOUT IT Because ft t and gt t are odd functions, what can be said about the function ht ftgt? 7. GRAPHICAL ANALYSIS With our graphing utilit in radian and parametric modes, enter the equations XT T and YT T and use the following settings. Tmin 0, Tma., 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 0 0 t t t Undef. 0 Undef. 0

8 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 insce, in Eercise 7 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.. 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. Hpotenuse Ug the lengths of these three sides, ou can form si ratios that define the si trigonometric functions of the acute angle. e ecant coe secant gent cogent Side adjacent to In the following definitions, it is import 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.) csc opp hp hp opp 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 opp adj cot adj opp

9 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 th 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 5 5. opp adj 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 5, 5, and 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. Ug the Pthagorean Theorem, ou find the length of the hpotenuse to be. 5 FIGURE.8 5 opp hp 5 adj hp 5 opp adj Now tr Eercise.

10 Section. Right Triangle Trigonometr 0 Eample Evaluating Trigonometric Functions of 0 and 0 Because the angles 0, 5, and 0,, 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 0, 0, and , FIGURE.9 0 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, opp 0 hp 0, and For adj, opp, and hp. So, opp 0 hp and Now tr Eercise 7. 0, 0 adj hp. 0 adj hp. Sines, Coes, and Tangents of Special Angles In the bo, note that 0 0. This occurs because 0 and 0 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 cot sec90 csc 90 cot90 csc90 sec

11 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 csc Quotient Identities Pthagorean Identities csc sec sec cot cot csc cot sec cot Note that represents, represents, and so on. Eample Appling Trigonometric Identities FIGURE Let be an acute angle such that Find the values of (a) and (b) ug trigonometric identities. a. To find the value of, use the Pthagorean identit. So, ou have 0. Substitute 0. for. Subtract 0. from each side. Etract the positive square root. b. Now, knowing the e and coe of, ou can find the gent of to be Use the definitions of and, and the triangle shown in Figure.0, to check these results. Now tr Eercise. 0..

12 Section. Right Triangle Trigonometr 0 Eample 5 Appling Trigonometric Identities Let be an acute angle such that Find the values of (a) cot and (b) sec ug trigonometric identities.. a. cot Reciprocal identit cot 0 FIGURE. b. sec Pthagorean identit sec sec 0 sec 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 e, coe, and gent to evaluate the ecant, secant, and cogent functions with a calculator. For insce, 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 insce, ou can find values of 8 and sec 8 as follows. Function Mode Calculator Kestrokes Displa a. 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, means the e of radian and means the e of degree. Eample Ug 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 5.7 Now tr Eercise 5. COS 5.7 ENTER

13 0 Chapter Trigonometr Applications Involving Right Triangles Object Observer Observer Angle of elevation Horizontal Horizontal Angle of depression 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 Object FIGURE. Ug Trigonometr to Solve a Right Triangle A surveor is sding 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.$. How tall is the Washington Monument? From Figure., ou can see that Angle of elevation 78. = 5 ft FIGURE 78.$ opp adj where 5 and is the height of the monument. So, the height of the Washington Monument is 78.$ feet. Not drawn to scale Now tr Eercise 7.. Eample 8 Ug 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 FIGURE 00 d. From Figure., ou can see that the e of the angle opp 00. hp 00 Now ou should recognize that 0$. Now tr Eercise 9. is

14 Section. Right Triangle Trigonometr 05 B now ou are able to recognize that is the acute angle that satisfies the equation Suppose, however, that ou were given the equation and were asked to find the acute angle. Because and 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 8. opp hp So, the length of the skateboard ramp is c c feet. Now tr Eercise 7.

15 0 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) Coe (c) Tangent (d) Cosecant (e) Secant (f) Cogent hpotenuse adjacent hpotenuse adjacent opposite (i) (ii) (iii) (iv) (v) (vi) adjacent opposite opposite hpotenuse hpotenuse 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 opposite adjacent 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 sec. 9. cot 0. csc In Eercises 0, construct an appropriate triangle to complete the table. 0 90, 0 /... sec. 5. cot. csc 7. csc 8. Function (deg) (rad) Function Value 9. cot sec

16 Section. Right Triangle Trigonometr 07 In Eercises, use the given function value(s), and trigonometric identities (including the cofunction identities), to find the indicated trigonometric functions (a) (c) (a) (c) (a) (c) (a) (c) (a) (c) (b) (b) (d) (b) (d) (b) (d) (b) (d) (b) (d) In Eercises 7, use trigonometric identities to transform the left side of the equation into the right side 0 < < / , sec 0 0 0, csc 0 0 sec 5 cot90 cot 5 (a) (c) cot90 7 sec cot cot sec 0 csc cot csc sec 0 (d) cot 0 0 cot 0 cot 0 In Eercises 7 5, use a calculator to evaluate each function. Round our answers to four decimal places. (Be sure the calculator is in the correct angle mode.) cot 0... cot. sec sec 7. (a) 0 (b) 80 csc90 csc (a).5 (b) cot.5 9. (a).5 (b) csc (a) cot 79.5 (b) sec (a) 50 5 (b) sec (a) sec (b) csc (a) cot 5 (b) 5 5. (a) sec (b) (a) csc 0 (b) 8 5. (a) sec 9 (b) cot In Eercises 57, find the values of in degrees 0 < < 90 and radians 0 < < / without the aid of a calculator. 57. (a) (b) csc 58. (a) (b) 59. (a) sec (b) cot 0. (a) (b). (a) csc (b). (a) cot (b) sec In Eercises, solve for,, or r as indicated.. Solve for.. Solve for Solve for.. Solve for r EMPIRE STATE BUILDING You are sding 5 meters from the base of the Empire State Building. You estimate that the angle of elevation to the top of the 8th floor (the observator) is 8. If the total height of the building is another meters above the 8th floor, what is the approimate height of the building? One of our friends is on the 8th floor. What is the disce between ou and our friend? r 0 0

17 08 Chapter Trigonometr 8. 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? 9. ANGLE OF ELEVATION You are skiing down a mountain with a vertical height of 500 feet. The disce 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 polluts 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 0 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. 0 5 (, ) mi (, ) 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. Ug a protractor, construct an angle of 0 in sdard 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 Not drawn to scale d 5 cm 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

18 Section. Right Triangle Trigonometr 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 decreag angle measures. (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 Angle, Height Angle, Height THINK ABOUT IT (a) Complete the table (b) Discuss the behavior of the e function for in the range from 0 to 90. (c) Discuss the behavior of the coe function for the range from 0 to 90. (d) Use the definitions of the e and coe functions to eplain the results of parts (b) and (c). 85. WRITING In right triangle trigonometr, eplain wh 0 regardless of the size of the triangle. 8. 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. 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 disce of 89.5 feet at an angle of approimatel 5., rig to a height of 9.5 feet above sea level. in TRUE OR FALSE? In Eercises 77 8, determine whether the statement is true or false. Justif our answer csc sec 0 csc cot 0 csc ft 9.5 feet above sea level Not drawn to scale 8. THINK ABOUT IT (a) Complete the table. (b) Is or greater for in the interval 0, 0.5? (c) As approaches 0, how do and compare? Eplain. (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.

19 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 insce, 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 sdard position with, a point on the terminal side of and r 0. sec r r (, ) cot, 0 r, 0, 0 csc r r, 0 Because r cannot be zero, it follows that the e and coe functions are defined for an real value of. However, if 0, the gent and secant of are undefined. For eample, the gent of 90 is undefined. Similarl, if 0, the cogent and ecant of are undefined. Eample Evaluating Trigonometric Functions James Urbach/SuperStock Let, be a point on the terminal side of. Find the e, coe, and gent of. Referring to Figure., ou can see that,, and r 5 5. So, ou have the following. r 5 r 5 (, ) r The formula r is a result of the Disce Formula. You can review the Disce Formula in Section.. Now tr Eercise 9. FIGURE.

20 Section. Trigonometric Functions of An Angle π < < π < 0 > 0 0 < < π > 0 > 0 The signs of the trigonometric functions in the four quadrants can be determined from the definitions of the functions. For insce, because it follows that 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 π < < π > 0 < 0 π < < π Given and > 0, find and 5 sec. Note that lies in Quadrant IV because that is the onl quadrant in which the gent is negative and the coe is positive. Moreover, ug Quadrant II : + : : Quadrant III : : : + FIGURE.7 Quadrant I : + : + : + Quadrant IV : : + : 5 and the fact that is negative in Quadrant IV, ou can let 5 and. So, r 5 and ou have r sec r.008. Now tr Eercise. Eample Trigonometric Functions of Quadrant Angles (, 0) FIGURE.8 π π (0, ) π 0 (0, ) (, 0) Evaluate the coe and gent 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. 0 r r 0 0 r r 0 0 Now tr Eercise undefined undefined,, 0, 0,,, 0, 0,

21 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 sdard 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: = π (radians) = 80 (degrees) FIGURE.9 Quadrant III = π (radians) = 80 (degrees) Quadrant IV = π (radians) = 0 (degrees) Eample Finding Reference Angles = 00 = 0 Find the reference angle. 00. a. b. c. 5 FIGURE.0 = π. 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

22 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 and So, it follows that and are equal, ecept possibl in sign. The same is true for and 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 hp r adj.. 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 e function that ma help ou remember the values Reverse the order to get coe values of the same angles. B ug reference angles and the special angles discussed in the preceding section, ou can greatl etend the scope of eact trigonometric values. For insce, 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 (degrees) (radians) Undef. 0 Undef.

23 Chapter Trigonometr Eample 5 Ug Reference Angles Evaluate each trigonometric function. a. b. 0 c. a. Because lies in Quadrant III, the reference angle is as shown in Figure.. Moreover, the coe 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 gent 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.. Because the ecant is positive in Quadrant II, ou have csc csc csc, = π = π = 0 = 0 = π = π FIGURE. FIGURE.5 FIGURE. Now tr Eercise 59.

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

25 Chapter Trigonometr. EXERCISES VOCABULARY: Fill in the blanks. See for worked-out solutions to odd-numbered eercises. In Eercises, let be an angle in sdard position, with, a point on the terminal side of and r sec 5.. r 7. Because r cannot be, the e and coe 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) (, ) In Eercises 8, the point is on the terminal side of an angle in sdard position. Determine the eact values of the si trigonometric functions of the angle.. 5,. 8, ,., , 7. 8., 7 ( 8, 5) (, ) (, ) (, ) In Eercises 9, state the quadrant in which 9. > 0 and 0. < 0 and. > 0 and. sec > 0 and cot lies. In Eercises, find the values of the si trigonometric functions of with the given constraint... Function Value Constraint 5. lies in Quadrant II.. lies in Quadrant III cot csc sec cot is undefined.. is undefined. In Eercises, 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 7 < 0 Quadrant. II. III 5. 0 III. 0 IV > 0 < 0 < 0 cot sec > 0 < 0 < 0 < 0 > 0

26 Section. Trigonometric Functions of An Angle 7 In Eercises 7, evaluate the trigonometric function of the quadrant angle In Eercises 5 5, find the reference angle, and sketch and in sdard position In Eercises 5 8, evaluate the e, coe, and gent of the angle without ug a calculator In Eercises 9 7, find the indicated trigonometric value in the specified quadrant. Function Quadrant Trigonometric Value 5 9. IV 70. cot II 7. III 7. csc IV I 9 7. sec III csc 9. sec 0. sec.. cot. csc. cot sec cot sec 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.) sec csc cot sec cot sec cot In Eercises 9 9, find two solutions of the equation. Give our answers in degrees 0 < 0 and in radians 0 <. Do not use a calculator. 9. (a) (b) 9. (a) (b) 9. (a) csc (b) cot 9. (a) sec (b) sec 95. (a) (b) cot 9. (a) (b) 97. DISTANCE An airplane, fling at an altitude of 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 disce d from the observer to the plane when (a) (b) and (c) 0. d 0, 90, mi Not drawn to scale 98. HARMONIC MOTION The displacement from equilibrium of an oscillating weight suspended b a spring is given b t t, 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