Practice Set Simplifying Trigonometric Expressions No Calculator Objectives Simplify trigonometric expression using the fundamental trigonometric identities Use the trigonometric co-function identities to evaluate trigonometric expressions involving a negative arc length. Derive and use the Pythagorean trigonometric identities to simplify trigonometric expressions. Notes The Fundamental Trigonometric Identities 1 1 sin cos 1 sec csc tan cot cot cos sin cos sin tan The Trigonometric Co-Function Identities cos cos sin sin The Pythagorean Trigonometric Identities sin cos 1 tan 1 sec 1 cot csc 1. (ACT/SAT) Multiple choice - calculator required An umbrella sprinkler is positioned on a ceiling at a point whose x-coordinate is 0. Negative values of x indicate distances, in meters, to the left of the position of the sprinkler, and positive values indicate distances to the right. s 1 The path of water from the sprinkler can be modeled by the quadratic function w x x 1 where w(x) is the height of the water, in meters, at position x. Which of the following equivalent expressions displays the height of the ceiling as a constant or coefficient? A. 1 x 1 x x 1 B. x 1 x 1 1 x x 1 C. D.. (ACT/SAT) Multiple choice calculator required Ariel was playing baseball and hit the ball into the air with a baseball bat. The height, h, in feet, of the ball t seconds after it left her bat is modeled by the equation h t 16t 6t. How many seconds after leaving Ariel s bat does the ball reach its maximum height? A. seconds B. seconds C. 8 seconds D. 16 seconds Trigonometric Investigations II 1
For problems -, find f ' x. You need NOT simplify your answer.. (1) f x x 1 x x. (1) fx x 5 x 5x 1 5. () Find the equation of the line tangent to f x x 1 x x at x = 1. 6. () A particle travels along the x-axis so that its position at any time t 0 is given by st t 15t t 7. For what values of t is the particle at rest? For problems 7-8, find the derivative of each function. 7. () y x 6x 1 8. () fx 1 x For problems 9-1, simplify each expression to a single trigonometric function or constant. 9. csc x cot x 10. tanx csc x sinx 11. csc x cot x sec x tan x Trigonometric Investigations II
1. sec x 1 1 cos x 1. sec xsin x 1. sec x sinx csc x cosx For problems 15-6, evaluate each of the following. 15. 7 sin 16. csc 17. 5 tan 18. 5 cot 6 19. cos 0. sec 1. tan. 7 sec. sin 6. cos 5. 5 csc 6 6. cot Trigonometric Investigations II
For problems 7-8, simplify each of the following to a single trigonometric function or constant. 7. 1 cos x 1 cos x 1 1 8. 9. csc x sec x 1 sin x 1 cosx 0. sec x tanx csc x 1 1. sec x 1 sec x. tan x 1 cos x 1 Trigonometric Investigations II
. 1 cos x1 cot x. cosx sinxtanx 5. cos xcot x 1 1 sinx csc x sinx 6. cot x 7. cot xcosxtanx sinx Trigonometric Investigations II 5
Practice Set 5 Rewriting Trigonometric Expressions No Calculator Objectives Show that a trigonometric expression can be rewritten in another specified equivalent format. Notes The Fundamental Trigonometric Identities 1 1 sin cos 1 sec csc tan cot cot cos sin cos sin tan The Trigonometric Co-Function Identities cos cos sin sin The Pythagorean Trigonometric Identities sin cos 1 tan 1 sec 1 cot csc 1. (ACT/SAT) Calculator required Amelia is a meteorologist measuring the movement of air at a warm front using an airborne sensor. She finds that the elevation, E, in meters of a particular volume of air t seconds after the start of recording is approximately E 0 0.08t 5. What was the elevation in meters of the volume of air at the start of recording? t. (ACT/SAT) Multiple choice calculator required The function Lt 5. 1.05 gives the approximate percent literacy rate in India t years after 1900. Which of the following equivalent functions shows, as a constant or coefficient, the approximate number of years it took for the literacy rate to triple? t t A. Lt 5..5 B. Lt 5. 1.077 t t C. Lt 5. 1.008 D. Lt 1.05 For problems -, find f ' x. You need NOT simplify your answer.. (1) f x x x 5x x. (1) fx 5x 1 x 5x 1 5. () Find the equation of the line tangent to fx x x 1 at x = 1. 6 Trigonometric Investigations II
6. () If s t t 1 t t 1, find the average acceleration of the object over the interval [0, ]. For problems 7-8, find the derivative of each function. 7. () x y x 8. () f x 5x x 7 sinx cosx 9. Show that 10. Show that cos x sin x csc x sec x can be rewritten as sinxcsc x. can be rewritten at 1 sin x. 11. Show that cos x sin x 1. Show that can be rewritten as sinx sinx cos x cos x 1. can be rewritten as tanx. 1 tanx Trigonometric Investigations II 7
cos x sin x sin x 1. Show that tan x sin x 1. Show that sinx can be rewritten as tan x sin x. can be rewritten as cot x cos x. tanx cot x cot x tanx 15. Show that 16. Show that tanx cot x sinxcos x can be rewritten as sin x cos x. can be rewritten as csc x sec x. 1 sin x 1sinx 1sinx 17. Show that 18. Show that 1 cot x 1sinx 1sinx can be rewritten as sin x cos x. can be rewritten as tanxsec x. 8 Trigonometric Investigations II
19. Show that tanx cot x 0. Show that cos x 1sin x can be rewritten as sec xcsc x. can be rewritten as sec x tanx. 1. Show that 1 cot x 1 cot x. Show that 1 sinx cos x can be rewritten as 1 cos x. can be rewritten as cos x. 1 sin x. Show that 1 sinx tanx cot x. Show that 1 sinx sinxcos x can be rewritten at sec x tan x can be rewritten as tan x cot x Trigonometric Investigations II 9
Practice Set 6 Double Angle Identities Split Calculator Objectives Use given Pythagorean value of a trigonometric function to find values of other trigonometric functions, including double angle functions. Show the work necessary to rewrite trigonometric expressions in another given format, for the sake of easier differentiation and integration. Notes sinx sin x cos x cos x sin x tan x cos x cos x 1 tanx 1 tan x 1 sin x 1. (ACT/SAT) Multiple choice calculator required A new cylindrical grain silo is being built to replace the old silo by enlarging its radius. The height, 15 meters (m), will stay the same. The approximate volume, in V x 15x 180x 50, where x is the additional length of m, of the new silo is given by the equation the new radius in meters. What is the approximate radius of the old silo? A. m B. 6 m C. 15 m D. 90 m. (ACT/SAT) Multiple choice calculator required Shriya began growing a colony of bacteria in a culture. 0.015t The function Pt 10 gives the population in millions of bacteria in the culture after growing for t minutes. Approximately how long (to the nearest minute) will it take for Shriya s bacteria colony to grow to 100 times its original population? A. 1 minutes B. 67 minutes C. minutes D. 10 minutes For problems -, find f ' x. You need NOT simplify your answer.. (1) fx x x x 1 5 x x 6x. (1) f x x 5 x 5x x 5x 5. () Find the equation of the line tangent to fx x 5x 6 x at x = 1. 6. () A particle travels along the x-axis so that its position at any time t 0 the initial velocity of the particle. s t t t 9. Find is given by 0 Trigonometric Investigations II
For problems 7-8, find the derivative of each function. 7. () y x 7 x x 5 8. () fx 1 x Calculator Required: For problems 9-11, if cos x and x, evaluate each of the following. 5 9. csc x 10. cosx 11. sinx Calculator Required: For problems 1-1, if 5 tan x and x, evaluate each of the following. 1 1. tanx 1. sec x 1. csc x Calculator Required: For problems 15-17, if 5 csc x and 0 x, evaluate each of the following. 15. sinx 16. cot x 17. cosx Calculator Required: For problems 18-0, if 15 sin x and x, evaluate each of the following. 17 18. csc x 19. cosx 0. tanx Trigonometric Investigations II 1
Calculator Required: For problems 1-, if 7 cot x and x, evaluate each of the following. 1. sinx. tanx. cosx Calculator Required: For problems -6, if 1 sec x and x, evaluate each of the following. 5. tanx 5. sinx 6. csc x Calculator Required: For problems 7-9, if cosx and 0 x, evaluate each of the following. 5 7. sinx 8. cot x 9. cosx Calculator Required: For problems 0-, if sin x and x, evaluate each of the following. 5 0. csc x 1. cosx. tanx csc x. Show that cos x can be rewritten as cscx.. Show that sec x sec x can be rewritten as secx. Trigonometric Investigations II
5. Show that sinx cos x 6. Show that cot x tanx can be rewritten as 1 sinx. can be rewritten as cscx. 7. Show that 1 cosx can be rewritten as cos x. 8. Show that 8sin xcos x can be rewritten as 1 cosx. 9. Show that sin xcsc x 0. Show that sin xcos x sin xcos x 1 can be rewritten as sec x. can be rewritten as 1 sinx. Trigonometric Investigations II
1 sec xcsc x cot x tan x. Show that cosx can be rewritten as cot x. can be rewritten as csc x. 1. Show that. Show that 1 cosx 1 cosx can be rewritten as tan x. cot x tanx. Show that cot x tanx can be rewritten as cosx. 5. Show that cosx 1 1 tan x 6. Show that sinx 1 tan x can be rewritten at sinx. can be rewritten as cosx. Trigonometric Investigations II
Practice Set 7 Derivatives of Trigonometric Functions No Calculator Objectives Compute, evaluate, and apply derivatives of trigonometric functions. Notes f x sin x f ' x cos x f x cos x f ' x sin x f x tan x f ' x sec x f x cot x f ' x csc x f x sec x f ' x sec x tan x f x csc x f ' x csc x cot x 1. (ACT/SAT) Multiple choice calculator required The Golden Gate Bridge is a suspension bridge that consists of two cables hung from two towers of equal height that are approximately 180 m apart. The approximate height of each cable above the ground, in meters, can be modeled by the function h x 0.00071 x 180x 15 where x is the distance in meters measured from the left tower. What is the approximate height of the towers? A. 60 m B. 15 m C. 0.00071 m D. 180 m. (ACT/SAT) Multiple choice calculator required The present value, (PV), of an investment is the amount that should be invested today at a specified interest rate in order to earn a certain amount at a future date. The amount desired is called the future value. Approximately how much should be invested today in a savings account that earns % interest compounded annually in order to have $500 in years? A. $515 B. $70 C. $85 D. $50 For problems -, find f ' x. You need NOT simplify your answer.. (1) fx x 7x 8 x. (1) f x x xx 5x 5. () Find the equation of the line tangent to fx x at x = 1. x Trigonometric Investigations II 5
6. () A particle travels along the x-axis so that its position at any time t 0 is given by s t t t 5t 6t 8. Find a(1). For problems 7-8, find the derivative of each function. 7. () x 5 y x 8. () f x x x x 1 For problems 9-17, find f ' x. 9. f x cos x sin x 10. f(x) sec x 11. f(x) tan5x 1. f(x) csc x 1. f x cos x 1. f(x) sinx cos x 16. fx 15. f(x) cot x tan x 17. 1 tan x cotx f(x) cosx 6 Trigonometric Investigations II
For problems 18-6, evaluate each of the derivatives at the given value. 11 Find f ' if f x cos x 6 18. Find f ' if f x tan x 19. 0. Find f ' 0 if f x sin x Find f ' if f x sec x 1. Find f ' if f x cos x. Find f ' if f x sec x 6. Find f ' if f x sinx. Find f ' if f x csc x 5. Find f ' if f x cosx 6. For problems 7-, find the linearization of each function at the given value of x. 7. f x tan x at x 8. f x 5 sec x at x 9. f x cosx at x 6 6 0. f x sinx at x 1. f x cot x at x. f x csc x at x Trigonometric Investigations II 7
For problems -, find f ' x.. f x sin x. f x tan x 5. f x sec x 6. f x sec x cot x 7. f x sin x cos x 8. f x tan x cot x 9. f x sec x 0. f x cot x 1. f x sin x. f x cos x sin x. f x tan x. f x cos 5x 8 Trigonometric Investigations II
For problems 5-50, evaluate each of the derivatives at the given value. 5. f x cos x at x 6. f x sin x csc x at x 7. f x tan x at x 1 7 8. f x sec x at x 9. f x csc x at x 50. f x sin x at x For problems 51-56, find dy in terms of x and y. dx 51. x tan y 5. x tan xy 0 5. sec x cot y y 5. x y sin y 55. cos y sin x 1 56. 1 tan y tan y xy Trigonometric Investigations II 9
Practice Set 8 Assessment 11 Review 75 Points Split Calculator 1. (ACT/SAT) Calculator required The Golden Years Senior Citizen Center uses a phone tree to announce when the center will be closed for poor weather. When each person receives a phone call, that person has a m list of three more people to call. The function 10 cm 1 approximates the total number of calls made after m minutes since the start of the phone tree. Approximately how many minutes will it take for the number of calls to reach 6?. (ACT/SAT) Calculator required A steel ball is traveling through water with a speed of s meters per second, F 0.5 0.00 s 50 s.5. At what speed in where s is positive. The drag force, F, in newtons (N) is meters per second does the ball have a force of 0.5N on it? Section 1 Polynomial, Product, and Quotient Rule (No Calculator) ( pts) Find the derivative of: o a polynomial function o a function consisting of the product of two polynomials o a function consisting of the quotient of two polynomials Find the second derivative of a polynomial function. For problems -, find f ' x. You need NOT simplify your answer.. fx x 9 x 16. f x x 7x 1 Section Applications of the Derivative (No Calculator) ( pts) Find the equation of the line tangent to f(x) at x = a. Find the equation of the line normal to f(x) at x = a. Apply the derivative to position, velocity, and acceleration. 5. Find the equation of the line tangent to fx x x at x = 1. x 1 0 Trigonometric Investigations II
6. A particle travels along the x-axis so that its position at any time t 0 average acceleration of the particle on [1, ]. is given by v t t 6t. Find the Section The Chain Rule (No Calculator) ( pts) Find the derivative of the composition of two functions f gx For problems 7-8, find the derivative of each function. 7. x 1 y x 8. f x x x 5 Section Simplifying Trigonometric Expressions (No Calculator) (1 pts) Simplify trigonometric expression using the fundamental trigonometric identities Use the trigonometric co-function identities to evaluate trigonometric expressions involving a negative arc length. Derive and use the Pythagorean trigonometric identities to simplify trigonometric expressions. For problems 9-1, simplify each expression to a single trigonometric function or constant. 9. 1sin xsec x tan x 10. sec x tan xcsc x 1 11. cos x tan x cot x tan x 1 1. cos x cot x csc x Trigonometric Investigations II 1
For problems 1-16, evaluate each of the following. 1. cos 1. 5 csc 6 15. tan 16. sin Section 5 Rewriting Trigonometric Expressions (No Calculator) (1 pts) Show that a trigonometric expression can be rewritten in another specified equivalent format. 17. Show that tan x cot x 18. Show that cos x cot x 1 sin x can be rewritten as sec x csc x. can be rewritten as csc x 1. 1 1 19. Show that 1sin x 1sin x can be rewritten as tan x sec x. 0. Show that can be rewritten as sec x tan x sin x cos x. Trigonometric Investigations II
1. Show that sec x 1 sin x 1 sinx can be rewritten as cos x. Show that csc x tan x cot x sec x. can be rewritten as sin x cos x. Section 6 Double Angle Identities (Split Calculator) (18 pts) Use given Pythagorean value of a trigonometric function to find values of other trigonometric functions, including double angle functions. Show the work necessary to rewrite trigonometric expressions in another given format, for the sake of easier differentiation and integration. Calculator Required: For problems -, if sin x and x, evaluate each of the following. 5. tanx. cosx Calculator Required: For problems 5-6, if 5 tan x and 0 x, evaluate each of the following. 1 5. sec x 6. sinx Trigonometric Investigations II
Calculator Required: For problems 7-8, if 5 sec x and x, evaluate each of the following. 7. cosx 8. tanx Calculator Required: For problems 9-0, if 8 cos x and x, evaluate each of the following. 17 9. cosx 0. tan x 1. Show that 1 cos x cot x can be rewritten as sinx.. Show that 1 cos x 1 cos x can be rewritten as tan x.. Show that cot x 1 cot x. Show that sin x csc x 1 can be rewritten as c ot x. can be rewritten as sec x. Trigonometric Investigations II
5. Show that 6sinxcos x tanx 1 6. Show that 1 sin x can be rewritten as sinx tanx. can be rewritten as cos x sin x. Section 7 Derivatives of Trigonometric Functions (No Calculator) (1 pts) Compute, evaluate, and apply derivatives of trigonometric functions. For problems 7-, find f ' x. tan x 7. f x sin x cos x 8. fx 1 tan x sin x 9. 0. f x 1 1 cos x f x cos x sin x n Trigonometric Investigations II 5
1 tan x 1. f x. f x csc 5x. f x cos x 1. f x cot x For problems 5-8, evaluate each of the derivatives at the given value. Find g' if g x cos x sinx 6 5. x f ' if f x cot 8 6. Find f ' if f x sin x 8 7. Find f ' if f x sinx cosx 8. Find 6 Trigonometric Investigations II
For problems 9-50, find the linearization of each function at the given value of x. 9. f x sec x at x 50. f x sinx at x 6 For problems 51-5, Find dy in terms of x and y. dx 51. xy sin y 5. x tan y x y Trigonometric Investigations II 7
Answers to Selected Exercises Practice Set Simplifying Trigonometric Expression P. 1 1. A. A. f ' x x x x x 1. f ' x dy 5. y x 1 6. 1, 7. x 6x 1 x 6 dx 8. f ' x 10. cosx 11. cot x 1. sec x 1. tanx 1. tanx 15. 18. 19. 0 0. 1. 1.. 8. 1 9. cos x 0. tanx 1. 6. sinx 7. cosx sin x. x 5x 1 x x 5x 1 6x 5 x 5 6 x 9. sec x 16. 1 1. 5. 6. 0 7. tan x. 1. sec x 5. csc x 17. sin x Practice Set 5 Rewriting Trigonometric Expressions P. 6 1.. A. f ' x x 8xx x x x x 5 5x 5x 1 x 55x 1 5 1. f ' x 5. y x 1 x 5x 1 8. f ' x 5x x 715x 9-. Solutions may vary. 6. 6 7. dy x 1 x 1 x dx x x Practice Set 6 Double Angle Identities P. 0 5 x x 1 x x 6x 5x x 6x x x 1 1. B. A. f ' x 5 x x 6x 5. f ' x 10x x x x 5x 9x x 5x x 5 5. y 1 1x 1 7. 11. xx x 5 x x 7 dy x 7 dx x x 5 x x 5 5 0 0. 161 8. 7 1. 10 119 1. 9. 57 65 6 65 1. 1 1. 6 57 5 0. 1. 1.. 7 5 1 5 57 65 15. 5. 7. 10 119 8. f ' x 16. 5. 6. 0 1 x 5 17. 10 169 7 5 9. 18. 6. 1 1-6. Solutions may vary. 5 17 15 10. 19. 7. 6 65 7 5 161 89 8 Trigonometric Investigations II
Practice Set 7 Derivatives of Trigonometric Functions P. 5 1. B. B. f ' x 5. y 1 1 x 1 6. 0 7. 9. sinx 10. 6sec x tanx 11. 1. cos x 15. x 7x x x 7x 8 x dy x 5 x 1 x 5 dx x x 1 csc x 16.. f ' x x x 5x x 5x x 8. f ' x x xx 1 x x 5 sec 5x 1. 1csc x cot x 1. 1sinx sec x 17. csc xcot x 18. 1 6 0. y 1 x 1. 1.. 6. 16. 1 5. 6. 6 7. y 1 x 8.. 8.. 5 y x 6 9. y 0 6 x sec x y 0x. sinx. tan x cot x csc x 9. 6 sec x tanx 0. tan x sec x. 15 sin10x 5. 50. 51. cos y 5. y 56. x csc y xy y sec xy 1 x sec 5. 6 sec x tan x 6. 19. 0. 1 y 0 6 x csc x cot x 7. 0 cot x csc x 1. 1sinx sinx. 8sin8x 9 sec x tan x 5. y csc y 6. 7. 1 8. 1 9. 16 9 5. x cos y 55. 1 cot x cot y Practice Set 8 Assessment 11 Review P. 0 1. 50..5. f ' x 6. 6 7. xx 16 x x 9 x 16 x 1 x 1 dy x 1 dx x x 10. cot x 11. csc x 1. 17-. Solutions will vary. 9. 161 89 0. 15 8 0. sinx 1. 5. 1 6. 51. y x cos y 5. sec x 1. 7. 7 5. f ' x 7x 1 7x 5. y 1 x 1 8. f ' x x x 5 x 9. cos x 1. 15. 16. 1 5. 1 1 6. 10 169 1-6. Solutions will vary 7. sinx 8. sec x. 5csc 5xcot 5x. 8sin8x. 7. 8. 1 9. y x 50. y cot y 7. 57 65 8. 6 57 sec x 9. sin x 1cot x csc x x 6 Trigonometric Investigations II 9
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