Section 7.1 Exercises

Transcription

1 Section 7.1 Solving Trigonometric Equations and Identities 109 Section 7.1 Eercises Find all solutions on the interval 0 sin 1. sin 1.. cos 1. cos Find all solutions 5. sin 1 9. cos 5 6. sin 10. 8cos 6 7. cos t 8. cos t sint 1. sin t 1 Find all solutions on the interval [0, ) 10sin cos 6cos sint 15costsint 15. csc sec 17. secsin sin tan sin sin sin 0. cos 1. sec 7. csc t. sin wsin w1 0. 8sin 6sin cos tcos t cos cos cos ( ) 15cos 8. 9sinw sin ( w) sin t cos t cos tan 6sin. tan 6cos 7sin 8 0 sin t cost t cos.. cos t 5 5. tan tan 5 6. tan 9 tan 0 7. sincossincos sin cossin cos tan sin 0 0. cos cot 1. tan t sec t. 1tanw tan w

2 110 Chapter 7 Section 7. Eercises Find an eact value for each of the following sin 195. cos(165 ). cos(5 ) 1. sin cos 6. cos 7. sin 8. sin Rewrite in terms of sin 6 sin and cos 10. sin cos 6 1. cos Simplify each epression 1. csc t 1. sec w 15. cot 16. tan Rewrite the product as a sum 16sin 16 sin cos6tcos6t 19. sin5cos 0. 10cos5sin 10 Rewrite the sum as a product cos 6t cos t 1.. cos6u cosu. sin sin 7. sin h sin h 5. Given sin a and cos 1 b, and a and b are both in the interval, cos a b a. Find sin a b b. Find 5 6. Given sin a and cos 1 b, and a and b are both in the interval 0, cos a b a. Find sin a b b. Find Solve each equation for all solutions sin cos 6 cos sin coscossin sin 1 sin 6 cos 11 cos 6 sin cos5 cos sin5 sin

3 Section 7. Addition and Subtraction Identities 111 Solve each equation for all solutions cos 5 cos 1.. sin 5 sin. cos6 cos sin. cos8 cos sin 5 Rewrite as a single function Asin( B C) sin 6cos sin 5cos 7. 5sin cos 8. sin5 cos5 Solve for the first two positive solutions 5sin cos sin cos 1. sin5cos. sin cos 1 Simplify sin 7t sin 5t. cos 7t cos 5t. t t t cos t sin 9 sin cos9 Prove the identity tan 1. tan 1 tan 1 tan t 5. tan t 1 tan t 6. cosabcosab cosacosb cosab 1tan atan b 7. cosab 1tan atan b tan ab sin acosasin bcosb 8. tan ab sin acosasin bcosb 9. sin absin abcosb cos( a) sin sin y tan y cos cosy cosa b tanatanb cosacosb y y y 5. cos cos cos sin

4 11 Chapter 7 Section 7. Eercises 1. If sin. If cos 1 and is in quadrant 1, then find eact values for (without solving for ) 8 tan a. sin b. cos c. and is in quadrant 1, then find eact values for (without solving for ) tan a. sin b. cos c. Simplify the epressions cos 8 sin (8 ) sin (17 ) cos 9 sin (9 ) sin8cos(8 ) 10. Solve for all solutions on the interval [0, ) 6sin t 9sin t 0 cos 7 1 cos 7 sin (7 ) cos 6 sin (6 ) 6sin 5 cos(5 ) sintcost cos9cos sin t cost 16. cost sin t 17. cos6cos sin sin 0 8cos 8cos 1 Use the double angle, half angle, or power reduction formula to rewrite without eponents 19. cos (5 ) 0. cos (6 ) 1. sin (8 ) sin. cos. sin. cos 5. If csc 7 sin and , then find eact values for (without solving for ) a. sin b. cos c. tan 6. If sec a. sin and , then find eact values for (without solving for ) b. cos c. tan

5 Section 7. Double Angle Identities 11 Prove the identity sin tcost 1 sin t sin 1 cos sin tan 9. sin 1 tan sincos 0. tan cos 1 1. cot tan cot sin. tan 1 cos 1 tan. cos 1 tan 1 cost cost. sin tcos t sin t 1 5. sin sin cos sin ( ) cos cos ( ) sin ( )cos 6.

6 11 Chapter 7 Section 7. Eercises Find a possible formula for the trigonometric function whose values are in the following tables y y The displacement ht (), in centimeters, of a mass suspended by a spring is modeled by the function ht 8sin(6 t), where t is measured in seconds. Find the amplitude, period, and frequency of this function.. The displacement ht ( ), in centimeters, of a mass suspended by a spring is modeled by the function ht 11sin(1 t), where t is measured in seconds. Find the amplitude, period, and frequency of this function.. A population of rabbits oscillates 19 above and below average during the year, hitting the lowest value in January. The average population starts at 650 rabbits and increases by 160 each year. Find an equation for the population, P, in terms of the months since January, t.. A population of deer oscillates 15 above and below average during the year, hitting the lowest value in January. The average population starts at 800 deer and increases by 110 each year. Find an equation for the population, P, in terms of the months since January, t. 5. A population of muskrats oscillates above and below average during the year, hitting the lowest value in January. The average population starts at 900 muskrats and increases by 7% each month. Find an equation for the population, P, in terms of the months since January, t. 6. A population of fish oscillates 0 above and below average during the year, hitting the lowest value in January. The average population starts at 800 fish and increases by % each month. Find an equation for the population, P, in terms of the months since January, t. 7. A spring is attached to the ceiling and pulled 10 cm down from equilibrium and released. The amplitude decreases by 15% each second. The spring oscillates 18 times each second. Find an equation for the distance, D the end of the spring is below equilibrium in terms of seconds, t.

7 Section 7. Modeling Changing Amplitude and Midline A spring is attached to the ceiling and pulled 7 cm down from equilibrium and released. The amplitude decreases by 11% each second. The spring oscillates 0 times each second. Find an equation for the distance, D the end of the spring is below equilibrium in terms of seconds, t. 9. A spring is attached to the ceiling and pulled 17 cm down from equilibrium and released. After seconds the amplitude has decreased to 1 cm. The spring oscillates 1 times each second. Find an equation for the distance, D the end of the spring is below equilibrium in terms of seconds, t. 10. A spring is attached to the ceiling and pulled 19 cm down from equilibrium and released. After seconds the amplitude has decreased to 1 cm. The spring oscillates 1 times each second. Find an equation for the distance, D the end of the spring is below equilibrium in terms of seconds, t. Match each equation form with one of the graphs ab sin 5 sin 5 m b 1. a. b. 1. a. sin 5 ab b. m bsin(5 ) I II III IV Find an equation of the form y ab csin y that fits the data given y Find an equation of the form y asin mb that fits the data given y y - 6 Find an equation of the form y 11 1 y ab cos c that fits the data given y

8 116 Chapter 7