Practice Set 44 – Simplifying Trigonometric Expressions No ... - Workbook.… · 4 − = − where w(x ... cosx sinx sin x. 3 ... 15 meters (m), will stay the same. The approximate

Trigonometric Investigations II 221

Practice Set 44 – 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 1sec csc tan cot cot

cos sin cos sin tan

The Trigonometric Co-Function Identities

cos cos sin sin

The Pythagorean Trigonometric Identities

2 2 2 2 2 2sin 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

The path of water from the sprinkler can be modeled by the quadratic function 21w x x 12

4

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. 21x 3

4

B. 1

x 12 x 124

C. 1

x 2 x 2 24

D.

1x 4 x 4 1

4

2. (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

2h t 16t 64t 4 . How many seconds after leaving Ariel’s bat does the ball reach its maximum height?

A. 2 seconds B. 4 seconds C. 8 seconds D. 16 seconds

222 Trigonometric Investigations II

For problems 3-4, find f ' x . You need NOT simplify your answer.

3. (31) 2f x 2x 1 x 3x 4 4. (31) 2

2

x 5f x

3x 5x 1

5. (32) Find the equation of the line tangent to 2f x 2x 1 x 3x 4 at x = 1.

6. (32) A particle travels along the x-axis so that its position at any time t 0 is given by

3 2s t 2t 15t 24t 7 . For what values of t is the particle at rest?

For problems 7-8, find the derivative of each function.

7. (34) 3

2y x 6x 1 8. (34) 3

1f x

2x 3

For problems 9-14, 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 tanx


12. sec x 1

1 cos x

13. sec x sin x 14.

sec x sinx

csc x cosx

For problems 15-26, evaluate each of the following.

15. 7

sin4

16.

4csc

3

17.

5tan

3

18. 5

cot6

19.

3cos

2

20.

2sec

3

21. 3

tan4

22.

7sec

4

23. sin

6

24. 2

cos3

25.

5csc

6

26. cot

2


For problems 27-38, simplify each of the following to a single trigonometric function or constant.

27. 4

2

1 cos x

1 cos x

28.

2 2

1 1

csc x sec x 29.

2sin x1

1 cosx

30. sec x tanx

csc x 1

31.

2

2

sec x 1

sec x

32. 2 2tan x 1 cos x 1


33. 2 21 cos x 1 cot x 34. cosx sinxtanx 35. cosxcot x

11 sinx

36. 2

csc x sinx

cot x

37. cot x cosx tanx sinx


Practice Set 45 – 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 1sec csc tan cot cot

cos sin cos sin tan

The Trigonometric Co-Function Identities

cos cos sin sin

The Pythagorean Trigonometric Identities

2 2 2 2 2 2sin 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 2

E 20 0.08 t 5 . What was the elevation in meters of the

volume of air at the start of recording?

2. (ACT/SAT) Multiple choice – calculator required – The function tL t 5.3 1.025 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?

A. t

44.5L t 5.3 3 B. t

3L t 5.3 1.077 C. 3tL t 5.3 1.008 D. t 23L t 3 1.025


3. (31) 3 2 4f x x 4x 5 x 2x 3 4. (31) 2

5x 1f x

2x 5x 1

5. (32) Find the equation of the line tangent to 2x 3

f xx 1

at x = 1.


6. (32) If 2s t t 1 t t 1 , find the average acceleration of the object over the interval [0, 2].


7. (34)

4x 3

yx 2

8. (34)

23f x 5x 2x 7

9. Show that sinx cosx

csc x sec x 10. Show that

2 2cos x sin x

can be rewritten as sinxcsc x . can be rewritten at 21 2sin x.

11. Show that 2 2cos x sin x 12. Show that

sinx

sinx cosx

can be rewritten as 22cos x 1 . can be rewritten as

tanx

1 tanx.


13. Show that 2 2tan x sin x 14. Show that

3cosx sinx sin x

sinx

can be rewritten as 2 2tan x sin x . can be rewritten as

2cot x cos x .

15. Show that tanx cot x

tanx cot x

16. Show that

cot x tanx

sinxcosx

can be rewritten as 2 2sin x cos x . can be rewritten as

2 2csc x sec x .

17. Show that 2

2

1 sin x

1 cot x

18. Show that

1 sinx 1 sinx

1 sinx 1 sinx

can be rewritten as 2 2sin xcos x . can be rewritten as 4tanxsec x .


19. Show that tanx cot x 20. Show that

cos x

1 sin x

can be rewritten as sec xcsc x . can be rewritten as sec x tanx .

21. Show that 2

2

1 cot x

1 cot x

22. Show that

1 sinx

cosx

can be rewritten as 21 2cos x . can be rewritten as

cos x.

1 sinx

23. Show that 1 sinx

1 sinx

24. Show that

tanx cot x

sinxcosx

can be rewritten at 2

sec x tanx can be rewritten as 2 2tan x cot x


Practice Set 46 – 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

2 2

2

2

2

cos x sin x2tanx

sin2x 2sinxcosx cos2x 2cos x 1 tan2x1 tan x

1 2sin 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

3m , of the new silo is given by the equation 2V x 15x 180x 540 , where x is the additional length of

the new radius in meters. What is the approximate radius of the old silo? A. 3 m B. 6 m C. 15 m D. 90 m 2. (ACT/SAT) Multiple choice – calculator required – Shriya began growing a colony of bacteria in a culture.

The function 0.015tP t 3 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. 133 minutes B. 67 minutes C. 33 minutes D. 10 minutes


3. (31) 3 2

5 2

x 2x x 1f x

x x 6x

4. (31) 5 4 3 2f x 2x x 5 3x 2x 5x

5. (32) Find the equation of the line tangent to 2x 5x 6

f x2x 3

at x = 1.

6. (32) A particle travels along the x-axis so that its position at any time t 0 is given by 3 2s t t 2t 9 . Find

the initial velocity of the particle.



7. (34)

42

2

x 7y

x 2x 5

8. (34)

41

f x4 3x

Calculator Required: For problems 9-11, if 3 3

cosx and x 25 2

, evaluate each of the following.

9. csc x 10. cos2x 11. sin2x


tanx and x12 2


12. tan2x 13. sec x 14. csc x

Calculator Required: For problems 15-17, if 5

csc x and 0 x4 2


15. sin2x 16. cot x 17. cos2x


sinx and x17 2


18. csc x 19. cos2x 20. tan2x



cot x and x 224 2


21. sin2x 22. tan2x 23. cos2x


sec x and x5 2


24. tan2x 25. sin2x 26. csc x


cosx and 0 x25 2


27. sin2x 28. cot x 29. cos2x


sinx and x 25 2


30. csc x 31. cos2x 32. tan2x

33. Show that csc x

2cosx 34. Show that

2

2

sec x

2 sec x

can be rewritten as csc2x . can be rewritten as sec2x.


35. Show that 2

sinx cosx 36. Show that cot x tanx

can be rewritten as 1 sin2x . can be rewritten as 2csc2x .

37. Show that 1 cos2x

2

38. Show that 2 28sin xcos x

can be rewritten as 2cos x . can be rewritten as 1 cos4x .

39. Show that 2 22sin xcsc 2x 40. Show that 32sinxcosx 4sin xcosx

can be rewritten as 21sec x

2

. can be rewritten as

1sin4x

2.


41. Show that 1

cot x tanx2

42. Show that sec xcsc x

cos2x

can be rewritten as cot2x . can be rewritten as 4csc4x .


1 cos2x

44. Show that

cot x tanx

cot x tanx

can be rewritten as 2tan x . can be rewritten as cos2x.

45. Show that cos2x 1

2sinx

46. Show that

2

2

1 tan x

1 tan x

can be rewritten at sinx . can be rewritten as cos2x .


Practice Set 47 – Derivatives of Trigonometric Functions No Calculator Objectives

Compute, evaluate, and apply derivatives of trigonometric functions. Notes

2 2

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 1280 m apart.

The approximate height of each cable above the ground, in meters, can be modeled by the function

2h x 0.000371 x 1280x 152 where x is the distance in meters measured from the left tower. What is

the approximate height of the towers? A. 640 m B. 152 m C. 0.000371 m D. 1280 m 2. (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 3% interest compounded annually in order to have $500 in 2 years?

A. $515 B. $470 C. $485 D. $530


3. (31) 2

3

x 7x 8f x

x 4

4. (31) 2 2f x x 3x x 5x 2

5. (32) Find the equation of the line tangent to 2x 3

f x3x 4

at x = 1.


6. (32) A particle travels along the x-axis so that its position at any time t 0 is given by

4 3 2s t 4t 3t 5t 6t 8 . Find a(1).


7. (34)

42x 5

yx 2

8. (34) 3 2f x x 2x 3x 1

For problems 9-17, find f ' x .

9. 2 2f x cos x sin x 10. f(x) 2sec 3x 11. f(x) tan5x

12. f(x) 3csc 4x 13. f x 3cos4x 14. f(x) 2sin2x cos2x

15. f(x) 4cot 3x 16. 2

2 tan xf x

1 tan x

17.

cot3xf(x)

cos3x


For problems 18-26, evaluate each of the derivatives at the given value.

18. 11

Find f ' if f x cos x6

19.

4Find f ' if f x tan x

3

20. Find f ' 0 if f x sinx

21. 3

Find f ' if f x sec x4

22. Find f ' if f x 3cos2x

4

23. Find f ' if f x 4sec 2x

6

24. 2

Find f ' if f x sin2x3

25. Find f ' if f x 3csc 2x

3

26. Find f ' if f x 2cos3x

2

For problems 27-32, find the linearization of each function at the given value of x.

27. f x tan x at x4

28.

5f x sec 2x at x

6

29. f x 2cos3x at x

6

30. 2

f x sin2x at x3

31.

3f x 3cot 2x at x

4

32.

3f x 2csc x at x

2



33. 2f x sin x 34. f x tan x 35. 3f x 2sec x

36. 2 2f x sec xcot x 37. 2 2f x sin x cos x 38. 3f x tanxcot x

39. 2f x sec 3x 40. 2f x cot 2x 41. 3f x 4sin 2x

42. 2 2f x cos 4x sin 4x 43. 3f x 2 tan 4x 44. 2f x 3cos 5x



45. 3f x cos 2x at x12

46. 3 2

f x sin 2x csc 2x at x3

47. 2 3

f x 3 tan x at x4

48. 2 7f x 3 sec x at x

4

49. 2f x csc 2x at x

3

50. 2f x sin 3x at x

4

For problems 51-56, find dy

dxin terms of x and y.

51. x tan y 52. x tan xy 0 53. 2sec x cot y y

54. 2x 2y sin y 55. cos2y sin x 12 56. 21 tan y

xy2 tan y


Practice Set 48 – 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

list of three more people to call. The function m

103

c m 3 12

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 363?

2. (ACT/SAT) Calculator required – A steel ball is traveling through water with a speed of s meters per second,

where s is positive. The drag force, F, in newtons (N) is F 0.5 0.004 s 50 s 2.5 . At what speed in

meters per second does the ball have a force of 0.5N on it? Section 31 – Polynomial, Product, and Quotient Rule (No Calculator) (3 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.


3. 2

2

x 9f x

x 16

4. f x 3x 2 7x 1

Section 32 – Applications of the Derivative (No Calculator) (3 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 2x x

f x2x 1

at x = 1.


6. A particle travels along the x-axis so that its position at any time t 0 is given by 2v t 3t 6t 4 . Find the

average acceleration of the particle on [1, 3]. Section 34 – The Chain Rule (No Calculator) (3 pts)

Find the derivative of the composition of two functions… f g x


7.

42x 1

yx 4

8.

33f x x 2x 5

Section 44 – Simplifying Trigonometric Expressions (No Calculator) (12 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 9-12, simplify each expression to a single trigonometric function or constant.

9. 1 sinx sec x tanx 10. sec x tanx csc x 1

11. cos x tan x cot x 12. 2

2

tan x 1cos x

cot x csc x


For problems 13-16, evaluate each of the following.

13. cos4

14.

5csc

6

15. 4

tan3

16.

3sin

2

Section 45 – Rewriting Trigonometric Expressions (No Calculator) (12 pts)

Show that a trigonometric expression can be rewritten in another specified equivalent format.

17. Show that tanx 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 .

19. Show that1 1

1 sin x 1 sin x

20. Show that

2 2 2sec x tan x sin x

can be rewritten as 2tanx sec x . can be rewritten as 2cos x .


21. Show that sec x

1 sin x 22. Show that

2 2csc x tan x cot x

sec x

can be rewritten as 3

1 sinx

cos x

. can be rewritten as sinx cos x .

Section 46 – 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.


sin x and x5 2


23. tan2x 24. cos2x


tan x and 0 x12 2


25. sec x 26. sin2x



sec x and x 224 2


27. cos2x 28. tan2x


cos x and x17 2


29. cos2x 30. tan x


cot x

32. Show that

1 cos2x

1 cos2x

can be rewritten as sin2x . can be rewritten as 2tan x .

33. Show that 2cot x 1

2cot x

34. Show that

2 22sin x csc 2x

can be rewritten as c ot 2x . can be rewritten as 21sec x

2

.


35. Show that 6sinxcosx tanx 36. Show that 211 sin 2x

2

can be rewritten as 3sin2x tanx . can be rewritten as 4 4cos x sin x .

Section 47 – Derivatives of Trigonometric Functions (No Calculator) (21 pts)

Compute, evaluate, and apply derivatives of trigonometric functions.


37. 2 2f x sin x cos x 38. 2

4 tan xf x

1 tan x

39. 2sin x

f x 11 cos x

40. 2 2f x cos 2x sin 2x

n


41. 1

f x tan2x2

42. f x csc 5x

43. 2f x 2cos 4x 1 44. 2f x 3cot 2x


45. Find g' if g x cos2x sin2x6

46. Find

3xf ' 2 if f x cot

8

47. Find 3f ' if f x 2sin 2x8

48. Find f ' if f x sin2x cos3x

3


For problems 49-50, find the linearization of each function at the given value of x.

49. f x sec x at x4

50. f x 3sin2x at x

6

For problems 51-52, Find dy

dxin terms of x and y.

51. 2xy sin y 2 52. x tan y 2x y


Answers to Selected Exercises

Practice Set 44 – Simplifying Trigonometric Expression – P. 221

1. A 2. A 3. 2f ' x 2 x 3x 4 2x 3 2x 1 4.

2 2

22

2x 3x 5x 1 6x 5 x 5f ' x

3x 5x 1

5. y 2 3 x 1 6. 1, 4 7. 2

2dy3 x 6x 1 2x 6

dx 8.

4

6f ' x

2x 3

9. sec x

10. cosx 11. 3cot x 12. sec x 13. tanx 14. 2tanx 15.

2

2 16.

2 3

3 17. 3

18. 3 19. 0 20. 2 21. 1 22. 2 23. 1

2

24.

1

2

25. 2 26. 0 27.

2sin x

28. 1 29. cos x 30. tanx 31. 2sin x 32.

2tan x 33. 1 34. sec x 35. csc x

36. sinx 37. 2cosx

Practice Set 45 – Rewriting Trigonometric Expressions – P. 226 1. 22 2. A 3. 2 4 3 3 2f ' x 3x 8x x 2x 3 4x 2 x 4x 5

4.

2

22

5 2x 5x 1 4x 5 5x 1f ' x

2x 5x 1

5.

5 1y x 1

2 4

6. 6 7.

3

2

1 x 2 1 x 3dy x 34

dx x 2 x 2

8. 3 2f ' x 2 5x 2x 7 15x 2 9-24. Solutions may vary.

Practice Set 46 – Double Angle Identities – P. 230

1. B 2. A 3.

2 5 2 4 3 2

25 2

3x 4x 1 x x 6x 5x 2x 6 x 2x x 1f ' x

x x 6x

4. 4 3 3 2 2 5 4f ' x 10x 4x 3x 2x 5x 9x 4x 5 2x x 5 5. y 12 31 x 1 6. 0

7.

3 2 22

2 22

2x x 2x 5 2x 2 x 7dy x 74

dx x 2x 5 x 2x 5

8.

5

12f ' x

4x 3

9.

5

4

10.

7

25

11. 24

25

12.

120

119 13.

13

12

14.

13

5

15.

24

25 16.

3

4 17.

7

25

18.

17

15

19.

161

289

20. 240

161

21.

336

625

22.

336

527 23.

527

625

24.

120

119 25.

120

169

26.

13

12 27.

336

625

28. 24

7 29.

527

625 30.

5

4

31.

7

25

32.

24

7 33-46. Solutions may vary.


Practice Set 47 – Derivatives of Trigonometric Functions – P. 235

1. B 2. B 3.

3 2 2

23

2x 7 x 4 3x x 7x 8f ' x

x 4

4. 2 2f ' x 2x 3 x 5x 2 2x 5 x 3x

5. y 1 1 x 1 6. 20 7.

3

2

2 x 2 1 2x 5dy 2x 54

dx x 2 x 2

8. 2 3 2f ' x 3x 4x 3x 1 3 x 2x

9. 2sin2x 10. 6sec 3x tan3x 11. 25 sec 5x 12. 12csc 4xcot 4x 13. 12sin4x

14. 4cos4x 15. 212csc 3x 16.

22 sec 2x 17. 3csc 3xcot 3x 18. 1

2 19. 4 20. 1

21. 2 22. –6 23. 16 3 24. –1 25. 4 26. –6 27. y 1 2 x4

28. 5

y 2 4 3 x6

29. y 0 6 x

6

30.

3 2y 1 x

2 3

31.

3y 0 6 x

4

32. 3

y 2 0 x2

33. sin2x 34.

2sec x

2 tan x 35.

36 sec x tan x 36. 22csc xcot x 37. 0

38. 22cot x csc x 39.

26 sec 3x tan3x 40. 24cot 2x csc 2x 41. 12sin2x sin4x 42. 8sin8x

43. 2 224 tan 4x sec 4x 44. 15sin10x 45.

9

4

46. 3 47. –12 48. –12 49.

16 3

9

50. –3 51. 2cos y 52.

2

2

y sec xy 1

x sec xy

53.

2

sec x tan x

2y csc y 54.

2x

2 cos y 55.

1cot x cot 2y

2

56. 2

y

x 2csc 2y

Practice Set 48 – Assessment 11 Review – P. 240

1. 50 2. 2.5 3.

2 2

22

2x x 16 2x x 9f ' x

x 16

4. f ' x 3 7x 1 7 3x 2 5. y 2 1 x 1

6. 6 7.

3

2

2 x 4 1 2x 1dy 2x 14

dx x 4 x 4

8. 2

3 2f ' x 3 x 2x 5 3x 2 9. cos x

10. cot x 11. csc x 12. 2sec x 13.

2

2 14. 2 15. 3 16. 1

17-22. Solutions will vary 23. 24

7

24.

7

25 25.

13

12 26.

120

169 27.

527

625 28.

336

527

29. 161

289

30.

15

8 31-36. Solutions will vary 37. 2sin2x 38.

24 sec 2x 39. sinx

40. 4sin4x 41. 2sec 2x 42. 5csc 5xcot 5x 43. 8sin8x 44.

212cot 2x csc 2x

45. 1 46. 3

4

47. 3 2 48. 1 49. y 2 2 x

4

50.

3 3y 3 x

2 6

51. 2y

2x cos y

52. 2cot y


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Practice Set 44 – Simplifying Trigonometric Expressions No ... - Workbook.… · 4 − = − where w(x ... cosx sinx sin x. 3 ... 15 meters (m), will stay the same. The approximate