Chapter Summary. What did you learn? 364 Chapter 4 Trigonometry

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1 0_00R.qd /7/0 : AM Page Chapter Trigonometr Chapter Summar What did ou learn? Section. Review Eercises Describe angles (p. 8)., Use radian measure (p. 8)., 8 Use degree measure (p. 8). 7 8 Use angles to model and solve real-life problems (p. 87). 9 Section. Identif a unit circle and describe its relationship to real numbers (p. 9). 8 Evaluate trigonometric functions using the unit circle (p. 9). 9 Use domain and period to evaluate sine and cosine functions (p. 97). Use a calculator to evaluate trigonometric functions (p. 98). 7 0 Section. Evaluate trigonometric functions of acute angles (p. 0). Use the fundamental trigonometric identities (p. 0). 8 Use a calculator to evaluate trigonometric functions (p. 0). 9 Use trigonometric functions to model and solve real-life problems (p. 0)., Section. Evaluate trigonometric functions of an angle (p. ) Use reference angles to evaluate trigonometric functions (p. ). 7 8 Evaluate trigonometric functions of real numbers (p. ) Section. Use amplitude and period to help sketch the graphs of sine 89 9 and cosine functions (p. ). Sketch translations of the graphs of sine and cosine functions (p. ). 9 9 Use sine and cosine functions to model real-life data (p. 7). 97, 98 Section. Sketch the graphs of tangent (p. ) and cotangent (p. ) functions Sketch the graphs of secant and cosecant functions (p. ). 0 0 Sketch the graphs of damped trigonometric functions (p. 7). 07, 08 Section.7 Evaluate and graph the inverse sine function (p. ). 09,, Evaluate and graph the other inverse trigonometric functions (p. ).,, Evaluate compositions of trigonometric functions (p. 7). 7 Section.8 Solve real-life problems involving right triangles (p. )., Solve real-life problems involving directional bearings (p. ). Solve real-life problems involving harmonic motion (p. ).

2 0_00R.qd /7/0 : AM Page Review Eercises Review Eercises. In Eercises and, estimate the angle to the nearest one-half radian... In Eercises 0, (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, convert the angle measure from degrees to radians. Round our answer to three decimal places º In Eercises 8, convert the angle measure from radians to degrees. Round our answer to three decimal places. 9. Arc Length Find the length of the arc on a circle with a radius of 0 inches intercepted b a central angle of Arc Length Find the length of the arc on a circle with a radius of meters intercepted b a central angle of 0.. Phonograph Compact discs have all but replaced phonograph records. 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?. Biccle At what speed is a bicclist traveling when his 7-inch-diameter tires are rotating at an angular speed of radians per second?. Circular Sector Find the area of the sector of a circle with a radius of 8 inches and central angle. Circular Sector Find the area of the sector of a circle with a radius of. millimeters and central angle In Eercises 8, find the point, on the unit circle that corresponds to the real number t.. t. t 7. t 8. t In Eercises 9, evaluate (if possible) the si trigonometric functions of the real number. 9. t 7 0. t. t. t In Eercises, evaluate the trigonometric function using its period as an aid.. sin. cos. sin 7. cos In Eercises 7 0, use a calculator to evaluate the trigonometric function. Round our answer to four decimal places. 7. tan 8. csc sec 0. sin. In Eercises, find the eact values of the si trigonometric functions of the angle shown in the figure

3 0_00R.qd /7/0 : AM Page Chapter Trigonometr In Eercises 8, 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 7. csc (a) sin (b) cos (c) sec (d) tan 8. csc (a) sin (b) cot (c) tan (d) sec90 In Eercises 9, use a calculator to evaluate the trigonometric function. Round our answer to four decimal places. 9. tan 0. csc. sin.. sec 79.. cot. cos Railroad Grade A train travels. kilometers on a straight track with a grade of 0 (see figure). What is the vertical rise of the train in that distance?. Gu Wire A gu wire runs from the ground to the top of a -foot telephone pole. The angle formed between the wire and the ground is. How far from the base of the pole is the wire attached to the ground?. In Eercises 7, 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. 7., 8., 9., 0. 0,. 0.,.. 0., 0..,, > 0.,, > 0. km 0 Not drawn to scale In Eercises 70, find the values of the si trigonometric functions of. Function Value. sec. csc 7. sin 8 8. tan 9. cos 70. sin Constraint In Eercises 7 7, find the reference angle, and sketch and in standard position In Eercises 7 8, evaluate the sine, cosine, and tangent of the angle without using a calculator. tan < 0 cos < 0 cos < 0 cos < 0 sin > 0 cos > In Eercises 8 88, use a calculator to evaluate the trigonometric function. Round our answer to four decimal places. 8. sin 8. tan 8. sin. 8. cot.8 8. sin 88. tan 7. In Eercises 89 9, sketch the graph of the function. Include two full periods. 89. sin 90. cos 9. f sin 9. f 8 cos 9. sin 9. cos 9. gt sint 9. gt cost 97. 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. (b) What is the frequenc of the sound wave described in part (a)?

4 0_00R.qd /7/0 : AM Page 7 Review Eercises Data Analsis: Meteorolog The times S of sunset (Greenwich Mean Time) at 0 north latitude on the th of each month are: (:9), (7:), (8:0), (8:8), (9:08), (9:0), 7(9:8), 8(8:7), 9(8:09), 0(7:), (:), (:). 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 (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.. In Eercises 99 0, sketch a graph of the function. Include two full periods. 0. f cot 0. gt cot t 0. f sec 0. ht sec t t St sin f tan 00. f t tan t f csc In Eercises 07 and 08, 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. 07. f cos 08. g cos.7 In Eercises 09, evaluate the epression. If necessar, round our answer to two decimal places. 09. arcsin 0. arcsin. arcsin 0.. arcsin 0.. sin 0.. sin 0.89 In Eercises 8, evaluate the epression without the aid of a calculator.. arccos. 7. cos 8. f t csc t arccos cos.8 In Eercises 9, use a calculator to evaluate the epression. Round our answer to two decimal places. 9. arccos arccos tan.. tan 8. In Eercises, use a graphing utilit to graph the function.. f arcsin. f arccos.. In Eercises 7 0, find the eact value of the epression. 7. cosarctan In Eercises and, write an algebraic epression that is equivalent to the epression.. f arctan f arcsin tanarccos secarctan cotarcsin tan arccos. secarcsin. Angle of Elevation The height of a radio transmission tower is 70 meters, and it casts a shadow of length 0 meters (see figure). Find the angle of elevation of the sun. 0 m 70 m. Height Your football has landed at the edge of the roof of our school building. When ou are 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 0 miles at a bearing of 8. From cit B to cit C, the plane flies 80 miles at a bearing of. Find the distance from cit A to cit C and the bearing from cit A to cit C.

5 0_00R.qd /7/0 : AM Page 8 8 Chapter Trigonometr. 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. 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. Snthesis True or False? In Eercises 7 0, determine whether the statement is true or false. Justif our answer. 7. The tangent function is often useful for modeling simple harmonic motion. 8. The inverse sine function arcsin cannot be defined as a function over an interval that is greater than the interval defined as. 9. sin is not a function because sin 0 sin Because tan, arctan. In Eercises, match the function a sin b with its graph. Base our selection solel on our interpretation of the constants a and b. Eplain our reasoning. [The graphs are labeled (a), (b), (c), and (d).] (a) (c) π π. Writing Describe the behavior of f sec at the zeros of g cos. Eplain our reasoning.. Conjecture (a) Use a graphing utilit to complete the table. (b) (d). sin. sin. sin. sin tan cot π (b) Make a conjecture about the relationship between tan 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 et0 cos t 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 to. (b) k is changed from 0 to. (c) b is changed from to Graphical Reasoning The formulas for the area of a circular sector and arc length are A r and s r, respectivel. ( r is the radius and is the angle measured in radians.) (a) For write the area and arc length as functions of r. What is the domain of each function? Use a graphing utilit to graph the functions. Use the graphs to determine which function changes more rapidl as r increases. Eplain. (b) For r 0 centimeters, write the area and arc length as functions of. What is the domain of each function? Use a graphing utilit to graph and identif the functions. 0. Writing Describe a real-life application that can be represented b a simple harmonic motion model and is different from an that ou ve seen in this chapter. Eplain which function ou would use to model our application and wh. Eplain how ou would determine the amplitude, period, and frequenc of the model for our application. 0.8, π t

6 0_00R.qd /7/0 : AM Page 9 Chapter Test 9 Chapter Test 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. (, ) 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 90 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 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.. Given that tan, find the other five trigonometric functions of.. Determine the reference angle of 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 < 0 if cos. (Do not use a calculator.) 9. Use a calculator to approimate two values of in radians 0 < if csc.00. Round the results to two decimal places. 90 In Eercises 0 and, find the remaining five trigonometric functions of the conditions. satisfing 0. cos, tan < 0. sec 7 8, sin > 0 f In Eercises and, sketch the graph of the function. (Include two full periods.). g sin. f tan π π π In Eercises and, use a graphing utilit to graph the function. If the function is periodic, find its period.. sin cos. e 0.t cos0.t, 0 t FIGURE FOR. 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 tanarccos without the aid of a calculator. 8. Graph the function f arcsin. 9. A plane is 80 miles south and 9 miles east of Cleveland Hopkins International 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 inches below equilibrium, bounces to its maimum height of inches above equilibrium, and returns to its lowest point in a total of seconds.

7 0_00R.qd /7/0 : AM Page 70 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 87. 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 70

8 0_00R.qd /7/0 : AM Page 7 P.S. 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. 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 : P.M. and was finished at 8:7 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 of teeth Number of teeth number in freewheel in chainwheel 0 9 Freewheel. A surveor in a helicopter is tring to determine the width of an island, as shown in the figure. 000 ft 7 9 d Chainwheel (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.. 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. If f is an even function and g is an odd function, use the results of Eercise 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 0 0 sin 8t B C where t is the time (in minutes). (The Ferris wheel has a radius of 0 feet.) This model ields a height of 0 feet when t 0. Alter the model so that the height of the car is foot when t 0. D E F G w Not drawn to scale 7

9 0_00R.qd /7/0 : AM Page 7 8. The pressure P (in millimeters of mercur) against the walls of the blood vessels of a patient is modeled b 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 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, 98. (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 00. (c) Calculate the person s three energ levels on September, (a) Use a graphing utilit to graph the functions given b 7 P 00 0 cos 8 t f cos sin and g cos sin. (b) Use the graphs from part (a) to find the period of each function. (c) If are positive integers, is the function given b and h A cos B sin P sin t, E sin t 8, periodic? Eplain our reasoning. t 0 t 0. Two trigonometric functions f and g have periods of, and their graphs intersect at.. (a) Give one smaller and one larger positive value of at which the functions have the same value. (b) Determine one negative value of at which the graphs intersect. (c) Is it true that f. g.? Eplain our reasoning.. The function f is periodic, with period c. So, f t c f t. Are the following equal? Eplain. (a) f t c f t (b) f t f t (c) 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). ft (a) You are standing in water that is feet deep and are looking at a rock at angle 0 (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 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?