C H A P T E R 4 Trigonometric Functions

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C H A P T E R 4

Trigonometric Functions

Section 4.1 Radian and Degree Measure Angles

1. Sketch a pair of coterminal angles. (a) x y (b) x y (c) x y (d) x y

2. Find the correct description of NP.

x y

P

N

M

q

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T-94 Chapter 4 Trigonometric Functions

3. Sketch an angle that is in standard position. (a) x y (b) x y (c) x y (d) x y Radian Measure

4. If possible, find the complement of the angleq= 4p 15. (a) 7 30 p _(b) p 15 (c) 11 15 p _{(d) not possible}

5. In which quadrant is the terminal side of the angleq ? q= -9p

10

(a) Quadrant I (b) Quadrant II (c) Quadrant III (d) Quadrant IV 6. Convert the measure to radians.

12

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Section 4.1 Radian and Degree Measure T-95

Degree Measure

7. In which quadrant is the terminal side of the angleq ?

q =105°

(a) Quadrant I (b) Quadrant II (c) Quadrant III (d) Quadrant IV

8. Classify the angle.

(a) obtuse (b) straight (c) right (d) acute

9. Find the complement of a 57° angle.

(a) 147° (b) 123° (c) 237° (d) 33°

Linear and Angular Speed

10. A bicycle wheel with a radius of 13 inches makes 2.1 revolutions per second. What is the speed of the bicycle?

(a) 1115 0. in. s (b) 1715. in. s (c) 3431. in. s (d) 54 6. in. s

11. A point on the rim of a wheel has a linear speed of 14 cm s. If the radius of the wheel is 20 cm, what is the angular speed of the wheel in radians per second?

(a) 1 4. rad s (b) 2 2. rad s (c) 0 7. rad s (d) 0 3. rad s

12. The needle of the scale in the bulk food section of a supermarket is 28 cm long. Find the distance the tip of the needle travels when it rotates 174°.

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Section 4.2 Trigonometric Functions: The Unit Circle The Unit Circle

13. A number line has its 0 fixed at 1 0

b g

, on a unit circle. What central angle, in radians, is formed by the radius to 1 0

b g

, and the radius to the point that corresponds to 0.76 on the number line when it is wrapped around the circle?

2 1 0 –1 –2 3 (1,0) (0, 1) (–1, 0) (0, –1) (a) 0 76. p (b) 43 54. p (c) 43.54 (d) 0.76

14. The point marking the bottom of a unit circle is positioned at 0 on a number line. The circle is rolled along the line until that point has rotated through an angle of 2

3

p _{. Find the number line location of the} point that marks the new location of the bottom of the circle.

(a) 114.59 (b) 2.09 (c) 1.33 (d) 0.67

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Section 4.2 Trigonometric Functions: The Unit Circle T-97

The Trigonometric Functions

16. Use the unit circle and a straightedge to approximate the value of the expression.

x

y

0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0 5.5 6.0 – .0 8 – .0 8 – .0 6 – .0 6 – .0 4 – .0 4 – .0 2 – .0 2 0 2. 0 2. 0 4. 0 4. 0 6. 0 6. 0 8. 0 8. cos .4 5 (a) 0 21. (b) 0 98. (c) –0.21 (d) –0.98

Use the unit circle and symmetry to help you evaluate the function(s).

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Use the unit circle and symmetry to help you evaluate the function(s).

I

KJ

0 1,

b g

18. csc

F

_HG

-7

I

_KJ

6 p (a) - 2 2 (b) - 3 (c) -2 3 3 (d) 2

Domain and Period of Sine and Cosine

19. Find an expression that completes the fundamental trigonometric identity. tan

b g

- =x

(a) cot x (b) - tan x (c) tan x (d) - cot x

20. Use the given trigonometric value to evaluate the indicated expression. If sint= 4, 19 find sin t

b

- 2p

g

. (a) 4 19 (b) -4 19 (c) 19 4 (d) 15 19

21. Use the period of the trigonometric function cosp

4 to evaluate the function.

(a) 2

2 (b) 0 (c) 1 (d)

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Section 4.2 Trigonometric Functions: The Unit Circle T-99

Evaluating Trigonometric Functions with a Calculator

22. Use a calculator to evaluate the function. (Be sure the calculator is in the correct angle mode.) sin .18 18°

(a) 3 2051. (b) –0.6206 (c) –0.7490 (d) 0 3120.

Use a calculator to evaluate the expression.

23. cos .0 8

(a) 1 4353. (b) 1 0001. (c) 0 9999. (d) 0 6967.

24. csc .6 1

(a) –5.4896 (b) –0.1822 (c) –1.4113 (d) –10.9792

Section 4.3 Right Triangle Trigonometry The Six Trigonometric Functions

25. Find the ratio a

b for the indicated angle and give its value.

45°

b

a c

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T-100 Chapter 4 Trigonometric Functions

26. Use the picture below to find the ratio that defines cot .q

b a c (a) a b (b) b a (c) b c (d) a c

27. Find the exact value of the sine and cosine functions of the angle q given in the figure. (Use the

Pythagorean Theorem to find the third side of the triangle.)

1

3

(a) sinq = 3 2 cosq = 1 2 (b) sinq = 3 3 cosq = 3 (c) sinq = 2 3 3 cosq = 3 2 (d) sinq = 1 2 cosq = 3 2 Trigonometric Identities

28. Use the fundamental trigonometric identities to determine the simplified form of the expression. tan

sin

ß ß

(a) csc ß (b) sec ß (c) cos ß (d) cot ß

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Section 4.3 Right Triangle Trigonometry T-101

Letq be an acute angle. Use the given function value and trigonometric identities to find the indicated trigonometric function.

30. If cscq = 26 find cot ., q

(a) 5 (b) 6 (c) 26 (d) 13

Evaluating Trigonometric Functions with a Calculator

31. Use a calculator to approximate cos13° (Round to the nearest four decimal places.).

(a) 0.1312 (b) 0.2250 (c) 0.2309 (d) 0.9744

32. Use a calculator to find the value of cos –

b

3215. .°

g

(Round to the nearest four decimal places.)

(a) 0 6225. (b) –0.6225 (c) –0.7826 (d) 0 7826.

33. Evaluate the expression using a calculator. cot .2 4

(a) –1.0917 (b) –2.1834 (c) –1.1397 (d) –0.9160

Applications Involving Right Triangles

34. A 12-foot ladder makes an angle of 50° with the ground as it leans against a house. How far up the house does the ladder reach?

(a) 9.19 ft (b) 15.66 ft (c) 14.30 ft (d) 7.71 ft

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T-102 Chapter 4 Trigonometric Functions

36. To find the height of a pole, a surveyor moves 100 feet away from the base of the pole and then, from an eye-level height of 6.5 feet, measures the angle of elevation to the top of the pole to be 40°. Find the height of the pole to the nearest foot.

(a) 90 ft (b) 71 ft (c) 83 ft (d) 84 ft

Section 4.4 Trigonometric Functions of Any Angle Introduction

37. Find the quadrant in whichq lies. tanq>0 and cosq<0

(a) Quadrant I (b) Quadrant II (c) Quadrant III (d) Quadrant IV

38. Given tanq= -12 sinq> ,

35 and 0 find cos .q (a) cosq =12 37 (b) cosq = -35 37 (c) cosq = -12 37 (d) cosq = 35 37 39. The point given is on the terminal side of an angle in standard position. Determine the exact value of

the sine of the angle. – ,9 40

b

g

(a) 40 1681 (b) 40 41 (c) -9 40 (d) -9 41 Reference Angles

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Section 4.4 Trigonometric Functions of Any Angle T-103

Trigonometric Functions of Real Numbers

43. Use a calculator to approximate two values ofq

b

0£ <q 2p

g

that satisfy the equation. cotq =1 055.

(a) 3.900, 1.544 (b) 3.900, 0.758 (c) 4.685, 1.544 (d) 4.685, 0.758 44. Find the exact value of the function.

cos 150°

(a) - 3

2 (b) - 2 (c) –1 (d)

3 3 45. Use a calculator to evaluate the trigonometric functions.

cot40° and sec .5 4

(a) 1.1918, 1.5756 (b) –0.8951, 1.5756 (c) –0.8951, 1.0045 (d) 1.1918, 1.0045 Section 4.5 Graphs of Sine and Cosine Functions

Basic Sine and Cosine Curves 46. Find the graph of the function.

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47. Find the function shown in the graph.

x y

–5 5

– 2p 2p

(a) y= sin .15x (b) y= 15. sinx (c) y= 15. cosx (d) y= cos .15x

48. Find the key points on the graph of the function.

KJ

Amplitude and Period of Sine and Cosine Curves

49. Find the amplitude and the period of the graphed function.

–5 –4 –3 –2 –1 1 2 3 4 5 - 2p 2p x y

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Section 4.5 Graphs of Sine and Cosine Functions T-105

50. Find a function that has the given amplitude and period. amplitude= 0 5. , period 8= p (a) y=0 25 x 4 . cos (b) y=0 5 x 4 . cos (c) y= 0 25. cos 8px (d) y=0 5 x 8 . cosp

51. Find the amplitude and the period of the function.

y=– . sin15 x 3 2 (a) Amplitude= –1.5 Period 3= p (b) Amplitude= 15. Period= 2 3 (c) Amplitude= –1.5 Period=2 3 (d) Amplitude= 15. Period 3= p

Translations of Sine and Cosine Curves

52. Graph the cosine function that has the given phase shift and vertical translation. phase shift= -p; vertical shift=

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Find the graph of the function.

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54. y=–4sin

F

_HG

x+p

I

_KJ

2 (a) x y –5 5 - p p (b) x y –5 5 - p p (c) x y –5 5 - p p (d) x y –5 5 - p p Mathematical Modeling

55. A weight attached to the end of a spring is pulled down 9 cm below its equilibrium point and released. It takes 11 seconds for it to complete one cycle of moving from 9 cm below the equilibrium point to 9 cm above the equilibrium point and then returning to its low point. Find the sinusoidal function that best represents the position of the moving weight and the approximate position of the weight

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56. The data below represent the average monthly cost of natural gas in a particular home.

Month Aug Sep Oct Nov Dec Jan

Cost ($) 20.70 27.03 44.83 67.45 92.07 108.37

Month Feb Mar Apr May Jun Jul

_KJ

+ 6 12 68 0 . sin p p .

57. A Ferris wheel has a radius of 19 feet and its center is 43 feet above the ground. The wheel rotates at a constant angular speed of 1

2

p _{per minute. The height of a point on the Ferris wheel as a function of}

time is given by h t

b g

=43 19+

F

_HG

1 t

I

_KJ

2

sin p , where h is the height in feet and t is the time in minutes. Find the approximate value of h t

b g

when t = 16 minutes.

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Section 4.6 Graphs of Other Trigonometric Functions T-109

Section 4.6 Graphs of Other Trigonometric Functions Graph of the Tangent Function

58. Find the function that is represented by the graph below.

–5 –4 –3 –2 –1 1 2 3 4 5 - 2p 2p x y (a) y=tan3x 2 (b) y x = -tan2 3 (c) y x = -cot 3 2 (d) y x =cot 2 3

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60. y= 3tan2x (a) –5 –4 –3 –2 –1 1 2 3 4 5 - 2p 2p x y (b) –5 –4 –3 –2 –1 1 2 3 4 5 - 2p 2p x y (c) –5 –4 –3 –2 –1 1 2 3 4 5 - 2p 2p x y (d) –5 –4 –3 –2 –1 1 2 3 4 5 - 2p 2p x y

Graph of the Cotangent Function

61. Find the function that is represented by the graph below.

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63. y=2 5 3x 4 . cot (a) –5 –4 –3 –2 –1 1 2 3 4 5 - 2p 2p x y (b) –5 –4 –3 –2 –1 1 2 3 4 5 - 2p 2p x y (c) –5 –4 –3 –2 –1 1 2 3 4 5 - 2p 2p x y (d) –5 –4 –3 –2 –1 1 2 3 4 5 - 2p 2p x y

Graphs of the Reciprocal Functions Sketch the graph of the function.

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Sketch the graph of the function.

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Sketch the graph of the function.

66. y= sec9x 4 (a) –5 –4 –3 –2 –1 1 2 3 4 5 - 2p 2p x y (b) –5 –4 –3 –2 –1 1 2 3 4 5 - 2p 2p x y (c) –5 –4 –3 –2 –1 1 2 3 4 5 - 2p 2p x y (d) –5 –4 –3 –2 –1 1 2 3 4 5 - 2p 2p x y

Damped Trigonometric Graphs 67. Graph y e x x

= - /2_cos

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T-118 Chapter 4 Trigonometric Functions (b) –5 –4 –3 –2 –1 1 2 3 4 5 - 2p 2px y (c) –5 –4 –3 –2 –1 1 2 3 4 5 - 2p 2px y (d) –5 –4 –3 –2 –1 1 2 3 4 5 - 2p 2_px y (69.)

Section 4.7 Inverse Trigonometric Functions Inverse Sine Function

70. Use a calculator to approximate the expression. arcsin .0 64

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Section 4.7 Inverse Trigonometric Functions T-119

71. If possible, evaluate the expression without the aid of a calculator. sin–1 3 2 (a) p 3 (b) p (c) p 2 (d) Not possible

72. Evaluate the expression without the aid of a calculator. arcsin

F

_HG

- 3

I

_KJ

2 (a) -p 4 (b) -p 6 (c) -p 3 (d) 0

Other Inverse Trigonometric Functions Use a calculator to approximate the expression.

73. arctan

b

- 0 03.

g

(a) –0.03 (b) –33.3233 (c) –0.0005 (d) –1909.86

74. arccos

b

- 0 34.

g

(a) 1.92 (b) 1.58 (c) 1.00 (d) 1.06

75. Evaluate the expression without the aid of a calculator. arctan 1 (a) p 3 (b) p 6 (c) – p 3 (d) p 4 Compositions of Functions

Use the properties of inverse functions to evaluate the expression.

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Use the properties of inverse functions to evaluate the expression.

77. csc arccos 4 5

F

HG

I

KJ

(a) 3 5 (b) 4 5 (c) 5 3 (d) 3 4 78. arcsin sin .

b

2 6

g

(a) 1.8464 (b) –0.5416 (c) 2.6 (d) 0.5416

Section 4.8 Applications and Models Applications Involving Right Triangles

79. An airplane is flying east at a constant altitude of 28,000 meters. When first seen to the east of an observer, the angle of elevation to the airplane is 71.5°. After 73 seconds, the angle of elevation is 51.6°. Find the speed of the airplane.

(a) 217 m s (b) 176 m s (c) 161 m s (d) 240 m s

80. At a distance of 56 feet from the base of a flag pole, the angle of elevation to the top of a flag that is 3.1 feet tall is 25.6°. The angle of elevation to the bottom of the flag is 22.9°. The pole extends 1 foot above the flag. Find the height of the pole.

(a) 26.8 ft (b) 24.8 ft (c) 23.8 ft (d) 27.8 ft

81. An energy company uses one wellhead to drill several exploratory wells at different angles. They strike oil when they have drilled 2879 feet along an angle of depression of 44°. Find the depth of the oil deposit.

Trigonometry and Bearings

82. A hiker travels at 3.9 miles per hour at a heading of S 21° E from a ranger station. After 3.5 hours, how far south and how far east is the hiker from the ranger station?

(a) 4 9. miles south and 12.7 miles east (b) 7 5. miles south and 11.4 miles east (c) 11 4. miles south and 7.5 miles east (d) 12 7. miles south and 4.9 miles east

83. A ship leaves port at 20 miles per hour, with a heading of S 44° W. There is a warning buoy located 5 miles directly north of the port. What is the bearing of the warning buoy as seen from the ship after 5.5 hours?

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Section 4.8 Applications and Models T-121

84. An airplane travels due east for 2.5 hours, at 400 miles per hour. Then it changes course to N 41° E. Find the airplane’s distance from its point of departure and its bearing, after a total flight time of 4.5 hours.

(a) Distance =1687 5. mi, at N 68.4 E° (b) Distance =1640mi, at N 21.6 E° (c) Distance =1640mi, at N 68.4 E° (d) Distance =1800mi, at N 21.6 E°

Harmonic Motion

85. A mass attached to a spring vibrates up and down in simple harmonic motion according to the equation h t

b g

_{= -}8 3t

4

sin where the amplitude, h, is in centimeters and the period, t, is in seconds. Find the amplitude and the period of the vibrations.

(a) amplitude cm period = sec = 8 3 4 (b) amplitude cm period = sec = 8 8 3 p (c) amplitude cm period = sec = 8 8 3 (d) amplitude cm period = sec = –8 3 4

86. A rowboat is observed from a dock as it bobs up and down in simple harmonic motion because of wave action. The boat moves from a high point of 3.3 feet below the dock to a low point of 8 feet below the dock and back to its high point 2 times every minute. Let t be time, in minutes, and h be the distance below the dock, in feet. Find an equation that describes the boat’s motion.

(a) h t

b g

=5 65 2 35. - . cos4pt (b) h t

b g

= -5 65 2 35. + . cospt

Trigonometric Functions (Answer Key)

Section 4.1 Radian and Degree Measure Angles [1] (c) [2] (c) [3] (c) Radian Measure [4] (a) [5] (c) [6] (d) Degree Measure [7] (b) [8] (d) [9] (d)

Linear and Angular Speed [10] (b)

[11] (c) [12] (d)

Section 4.2 Trigonometric Functions: The Unit Circle

The Unit Circle [13] (d) [14] (b) [15] (b)

The Trigonometric Functions [16] (c)

[17] (d) [18] (d)

Domain and Period of Sine and Cosine [19] (b)

[20] (a) [21] (a)

Evaluating Trigonometric Functions with a Calculator

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Section 4.3 Right Triangle Trigonometry The Six Trigonometric Functions

[25] (a) [26] (b) [27] (d) Trigonometric Identities [28] (b) [29] (c) [30] (a)

Evaluating Trigonometric Functions with a Calculator

[31] (d) [32] (d) [33] (a)

Applications Involving Right Triangles [34] (a)

[35] (b) [36] (a)

Section 4.4 Trigonometric Functions of Any Angle Introduction [37] (c) [38] (b) [39] (b) Reference Angles [40] (d) [41] (d) [42] (d)

Trigonometric Functions of Real Numbers [43] (b)

[44] (a) [45] (a)

Section 4.5 Graphs of Sine and Cosine Functions

Basic Sine and Cosine Curves [46] (a)

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Amplitude and Period of Sine and Cosine Curves

[49] (d) [50] (b) [51] (d)

Translations of Sine and Cosine Curves [52] (c) [53] (d) [54] (a) Mathematical Modeling [55] (c) [56] (c) [57] (b)

Section 4.6 Graphs of Other Trigonometric Functions

Graph of the Tangent Function [58] (a)

[59] (b) [60] (d)

Graph of the Cotangent Function

[62] (b) [63] (c)

Graphs of the Reciprocal Functions [64] (d) [65] (d) [66] (c) [67] (d) [68] (b) [69] (d)

Section 4.7 Inverse Trigonometric Functions

Inverse Sine Function [70] (a)

[71] (a) [72] (c)

Other Inverse Trigonometric Functions [73] (a)

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Compositions of Functions [76] (a)

[77] (c) [78] (d)

Section 4.8 Applications and Models Applications Involving Right Triangles [79] (b)

[80] (d) [81] (c)