PART I: NO CALCULATOR (144 points)

Transcription

1 Math 10 Practice Final Trigonometry 11 th edition Lial, Hornsby, Schneider, and Daniels (Ch. 1-8) PART I: NO CALCULATOR (1 points) (.1,.,.,.) For the following functions: a) Find the amplitude, the period, any vertical translation, and any phase shift. If not applicable, write none in the blank. b) Graph over the interval x. Identify and label any asymptotes y sin x. y cos x+ amplitude: amplitude: : period: vertical translation: phase shift: period: vertical translation: phase shift:. y tan x amplitude: period: vertical translation: phase shift:. y csc x amplitude: period: vertical translation: phase shift:

2 Math 10 Practice Final (cont.) (.1,.,.,.) For the following functions: a) Find the amplitude, the period, any vertical translation, and any phase shift. If not applicable, write none in the blank. b) Graph over the interval x. Identify and label any asymptotes y sec x 6. y cot x amplitude: period: vertical translation: phase shift: amplitude: period: vertical translation: phase shift: (6.1) Give the exact radian measure of y if it exists. 7. y arctan ( ) 8. y cos y sec ( ) 10. y arcsin y csc 1 1. y cot ( 1) Write the following trigonometric expression as an algebraic expression in u, for u > cot ( sec u) 1. cos( arcsin u )

3 Math 10 Practice Final (cont.) PART II: YOU MAY USE A CALCULATOR (56 points) sin A sin Acos A DOUBLE-ANGLE IDENTITIES cos A cos A sin A tan A tan A 1 tan A cos A cos A 1 cos A 1 sin A SUM AND DIFFERENCE IDENTITIES sin( A+ B) sin Acos B+ cos Asin B sin( A B) sin Acos B cos Asin B cos( A + B) cos Acos B sin Asin B cos( A B) cos Acos B + sin Asin B tan A + tan B tan( A + B) 1 tan Atan B tan A tan B tan( A B) tan Atan B LAW OF COSINES a b + c bccos A b a + c accos B c a + b abcosc LAW OF SINES a b c sin A sin B sinc HALF-ANGLE IDENTITIES A 1 cos A sin ± A cos A cos ± A sin A tan cos A A 1 cos A tan sin A A 1 cos A tan ± 1 + cos A DE MOIVRE S THEOREM ( cos sin ) n n r θ + i θ r ( cos nθ + isin nθ) where r( + i) is a complex number and n is any real number. sin A sin B sinc a b c

4 Math 10 Practice Final (cont.) (1.1) 1. Convert the following angles to decimal degrees. If applicable, round to the nearest hundredth of a degree. a) 76 8' b) 51'5'' c) 9 15' (1.). Identify the quadrant satisfying the given conditions: a) < 0 and cotθ > 0 b) tanθ < 0 and cscθ > 0 (.1,.). Find the exact values of the six trigonometric functions for the given angles. Rationalize denominators when applicable. a) 15 b) 10 c) 70 d) 00 (5.5). Given 5 cos x and 90 < x < 180, find the exact values of the following. 1 sin x cos x tan x 5. Given 7 5 csc x and cos x > 0, find the exact values of the following. 5 sin x cos x tan x (.5) 6. Find h as indicated in the figure. h m

5 Math 10 Practice Final (cont.) (7.) Solve the following problem. Include a labeled sketch in your work. 7. Starting at point A, a ship sails 18. km on a bearing of 190, then turns and sails 7. km on a bearing of 19. Find the distance of the ship from point A to the nearest tenth of a kilometer. (7.1) Solve the following problem. Include a labeled sketch in your work. 8. Francesco is flying in a hot air balloon directly above a straight road. mi long that joins two towns. He finds that the town closer to him is at an angle of depression of 5.9, and the farther town is at an angle of depression of 8.5. How high above the ground is the balloon? Round your answer to the nearest hundredth of a mile. (5.6) 9. Use the appropriate half-angle identity to find the exact value of the following. Work demonstrating use of appropriate identity must be shown. a) sin 0.5 b) tan 75 (5., 5.) 10. Use the appropriate sum or difference identity to find the exact value of the following. Work demonstrating use of appropriate identity must be shown. a) 5 cos 1 b) 1 tan 1 c) sin 1 (.1) 11. Convert the following angles to radians. Leave answers as multiples of. a) 110 b) 16 (.1) 1. Convert the following angles to degrees. a) 15 b) 8 5 5

6 Math 10 Practice Final (cont.) (5.1, 5., 5.5) Verify that each equation is an identity. sin tan 1. θ + θ tan θ 1. sec θcsc θ sec θ + csc θ 15. cos θ cotθ tanθ 16. tan 8 tan 8 tan tan θ θ θ θ (6.) 17. Solve the following equations for all exact solutions in radians. Write answers using the least possible nonnegative angle measures. a) sin x 0 b) cos x+ 1 sin x (6.) 18. Solve the following for all solutions in degrees. Use exact values for x whenever possible. If necessary, approximate answers to the nearest tenth of a degree. Write answers using the least possible nonnegative angle measures. a) cot x b) sin x sin x (6., 6.) 19. Solve the following equation over the interval 0, 60 ) Use exact values for x whenever possible. If necessary, approximate answers to the nearest tenth of a degree. a) 5 tan x+ 16 tan x 0 b) 5sec x + sec x c) sin x 1 (8.) 0. Use DeMoivre s Theorem to find ( ) 6 i. Write your answer in rectangular form. (8.5) 1. Convert the following to polar coordinates with 0 θ < 60 and r > 0. a) ( 1, ) b) (, ) 6

7 Math 10 Practice Final (cont.) (8.6). A golf ball is hit from the ground with initial velocity of 150 feet per second at an angle of 60 with the ground. x 75 t The parametric equations that model the path of the rocket are given by y 16t + 75 t Determine a rectangular equation that models the path of the projectile. Use exact values for any numbers in the equation. 7

8 Math 10 Practice Final (cont.) Part I Answers: 1) a) amplitude: vertical translation: none period: phase shift: none b) ) a) amplitude: vertical translation: none period: phase shift: left b) 8

9 Math 10 Practice Final (cont.) Part I Answers: ) a) amplitude: none vertical translation: none period: phase shift: right b) asymptotes: 5 x x x x 7 ) a) amplitude: none vertical translation: none period: phase shift: none b) asymptotes: x x x 0 x x 9

10 Math 10 Practice Final (cont.) Part I Answers: 5) a) amplitude: none vertical translation: up period: phase shift: none b) asymptotes: x x x x 6) a) amplitude: none vertical translation: none period: phase shift: none b) asymptotes: x x 0 x 10

11 Math 10 Practice Final (cont.) Part I Answers: 7) 8) 5 6 9) does not exist 10) 11) does not exist 1) 1) u 1 u 1 1) 1 u 11

12 Math 10 Practice Final (cont.) Part II Answers: 1) a) 78.8 b).86 c) 9.5 ) a) III b) II a) sin15 cos15 tan15 1 csc15 sec15 cot15 1 b) sin 10 1 cos 10 tan 10 csc 10 sec 10 cot 10 c) sin 70 1 cos 70 0 d) sin 00 cos 00 1 tan 00 csc 00 sec 00 cot 00 ) sin x 10 1 cos x 1 tan x

13 Math 10 Practice Final (cont.) Part II Answers: 5) 55 sin x 9 9 cos x 9 55 tan x 9 6) 8 m 7) 8.6 km 8) 1.1 miles 9) a) b) 10) a) 6 b) c) 6 11) a) b) 6 5 1) a) 8 b) 88 1

14 Math 10 Practice Final (cont.) Part II Answers: 1) One possible way of verifying + tanθ tanθ : + tanθ + 1 cos θ ( + 1) tanθ 1) One possible way of verifying sec θcsc θ sec θ + csc θ : sec θ + csc θ cos θ sin θ sin θ + cos θ sin θcos θ 1 sin θcos θ sec θcsc θ 1

15 Math 10 Practice Final (cont.) Part II Answers: 15) One possible way of verifying cos θ cotθ tanθ : cos θ cos θ sin θ cos θ sin θ cotθ tanθ 16) One possible way of verifying tan 8θ tan 8θ tan θ tan θ : tan 8 tan 8 tan θ θ θ tan 8θ tan tan θ 1 tan θ tan θ ( 1 tan θ ) ( θ )( 1 tan θ ) ( 1 tan θ ) 1 15

16 Math 10 Practice Final (cont.) Part II Answers: 17) a) + k + k, where k is an integer + k b) + k 5 + k, where k is an integer 18) a) k, where k is an integer b) k k, where k is an integer 19) a) 58.8,101.7, 8.8, 81.7 b) 7.8,. c) 5,15, 5, 15 0) 096 or i 1) a) (,10 ) b) (, 15 ) ) y x + x 16