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### Transcript of Chapter 5 Trigonometric Functions - · PDF file 2019. 3. 4. · 454 Chapter 5...

• Copyright © 2015 Pearson Education Inc. 451

Chapter 5 Trigonometric Functions

5.1 Angles and Their Measure

5.1 Practice Problems

1.

2. a. 9 22

13 9 22 13 13.16 60 3600

⎛ ⎞° = + + ° ≈ °′ ′′ ⎜ ⎟⎝ ⎠

b. 41.275 41 0.275(60 ) 41 16.5 41 16 0.5(60 ) 41 16 30

° = ° + = ° +′ ′ = ° + + = °′ ′′ ′ ′′

3. 45 45 radians 180 4

π π− ° = − ⋅ = −

4. 3 3 180

270 2 2

π π π

= ⋅ = °

5. complement: 90º − 67° = 23º supplement: 180º − 67° = 113º

6. First convert the angle measurement from degrees to radians:

5 225 225 radians

180 4

π π⎛ ⎞° = ° =⎜ ⎟⎝ ⎠° .

Then, 5 5

2 7.85 m 4 2

s r π πθ ⎛ ⎞= = = ≈⎜ ⎟⎝ ⎠

7. The difference in the latitudes is 41 51 30 25 11 26

26 11 11.43

60

11.43 180

π

° − ° = °′ ′ ′ ⎛ ⎞= ° + ° ≈ °⎜ ⎟⎝ ⎠ ⎛ ⎞= ° ⎜ ⎟⎝ ⎠°

0.1995 3960 790= ⋅ ≈s miles.

8. First convert the angle measurement from

degrees to radians: 60 radians 180 3

π πθ = ° ⋅ = °

.

2 2 21 1 5010 52.36 in. 2 2 3 3

π πθ= = ⋅ ⋅ = ≈A r

9. First convert revolutions per minute into radians per minute: 18 revolutions per minute = 18 2 36π π⋅ = radians per minute. Thus, the angular speed

36ω π= radians per minute. To find the linear speed, use the formula

:v rω= 10 36 360π π= ⋅ =v radians per minute or about 1131 feet per minute.

5.1 Basic Concepts and Skills

1. A negative angle is formed by rotating the initial side in the clockwise direction.

2. An angle is in standard position if its vertex is at the origin of a coordinate system and its initial side lie on the positive x-axis.

3. One second (1′′) is one-sixtieth of a minute.

4. False. One radian 180

57.3 π

= ° ≈ ° .

5. True. The circumference of the circle 2 2 1 2 6.28 m.π π π= = ⋅ = ≈r

6. True. Acute angles measure more than 0º and less than 90º.

7.

8.

• 452 Chapter 5 Trigonometric Functions

9.

10.

11.

12.

13.

14.

15. 45

70 45 70 70.75 60

⎛ ⎞° = + ° = °′ ⎜ ⎟⎝ ⎠

16. 38

38 38 38 38.63 60

⎛ ⎞° = + ° ≈ °′ ⎜ ⎟⎝ ⎠

17. 42 30

23 42 30 23 23.71 60 3600

⎛ ⎞° = + + ° ≈ °′ ′′ ⎜ ⎟⎝ ⎠

18. 50 50

45 50 50 45 45.85 60 3600

⎛ ⎞° = + + ° ≈ °′ ′′ ⎜ ⎟⎝ ⎠

19. 42 57

15 42 57 15 15.72 60 3600

⎛ ⎞− ° = − + + ° ≈ − °′ ′′ ⎜ ⎟⎝ ⎠

20. 18 13

70 18 13 70 70.30 60 3600

⎛ ⎞− ° = − + + ° ≈ − °′ ′′ ⎜ ⎟⎝ ⎠

21. 27.32 27 0.32(60 ) 27 19.2 27 19 0.2(60 ) 27 19 12

° = ° + = ° +′ ′ = ° + + = °′ ′′ ′ ′′

22. 120.64 120 0.64(60 ) 120 38.4 120 38 0.4(60 ) 120 38 24

° = ° + = ° +′ ′ = ° + + = °′ ′′ ′ ′′

23. 13.347 13 0.347(60 ) 13 20.82 13 20 0.82(60 ) 13 20 49

° = ° + = ° +′ ′ = ° + + = °′ ′′ ′ ′′

24. 110.433 110 0.433(60 ) 110 25.98 110 25 0.98(60 ) 110 25 59

° = ° + = ° +′ ′ = ° + + = °′ ′′ ′ ′′

25. 19.0511 19 0.0511(60 ) 19 3.066 19 3 0.066(60 ) 19 3 4

° = ° + = ° +′ ′ = ° + + = °′ ′′ ′ ′′

26. 82.7272 82 0.7272(60 ) 82 43.632 82 43 0.632(60 ) 82 43 38

° = ° + = ° +′ ′ = ° + + = °′ ′′ ′ ′′

27. 20 20 radian 180 9

π π° = ⋅ =

28. 2

40 40 radian 180 9

π π° = ⋅ =

29. 180 180 radian 180

π π− ° = − ⋅ = −

30. 7

210 210 radians 180 6

π π− ° = − ⋅ = −

31. 7

315 315 radians 180 4

π π° = ⋅ =

32. 11

330 330 radians 180 6

π π° = ⋅ =

33. 8

480 480 radians 180 3

π π° = ⋅ =

• Section 5.1 Angles and Their Measure 453

34. 5

450 450 radians 180 2

π π° = ⋅ =

35. 17

510 510 radians 180 6

π π− ° = − ⋅ = −

36. 7

420 420 radians 180 3

π π− ° = − ⋅ = −

37. 180

15 12 12

π π π

= ⋅ = °

38. 3 3 180

67.5 8 8

π π π

= ⋅ = °

39. 5 5 180

100 9 9

π π π

− = − ⋅ = − °

40. 3 3 180

54 10 10

π π π

− = − ⋅ = − °

41. 5 5 180

300 3 3

π π π

= ⋅ = °

42. 11 11 180

330 6 6

π π π

= ⋅ = °

43. 5 5 180

450 2 2

π π π

= ⋅ = °

44. 17 17 180

510 6 6

π π π

= ⋅ = °

45. 11 11 180

495 4 4

π π π

− = − ⋅ = − °

46. 7 7 180

420 3 3

π π π

− = − ⋅ = − °

For exercises 47−50, make sure your calculator is in Radian mode.

47. 12 12 0.21 radian 180

π⎛ ⎞° = ° ≈⎜ ⎟⎝ ⎠°

48. 127 127 2.22 radians 180

π⎛ ⎞° = ° ≈⎜ ⎟⎝ ⎠°

49. 84 84 1.47 radians 180

π⎛ ⎞− ° = − ° ≈ −⎜ ⎟⎝ ⎠°

50. 175 175 3.05 radians 180

π⎛ ⎞− ° = − ° ≈ −⎜ ⎟⎝ ⎠°

For exercises 51−54, make sure your calculator is in Degree mode.

51. 180

0.94 radians 0.94 53.86 π °⎛ ⎞= ≈ °⎜ ⎟⎝ ⎠

52. 180

5 radians 5 286.48 π °⎛ ⎞= ≈ °⎜ ⎟⎝ ⎠

53. 180

8.21 radians 8.21 470.40 π °⎛ ⎞− = − ≈ − °⎜ ⎟⎝ ⎠

• 454 Chapter 5 Trigonometric Functions

54. 180

6.28 radians 6.28 359.82 π °⎛ ⎞− = − ≈ − °⎜ ⎟⎝ ⎠

55. complement: 43°; supplement: 133°

56. complement: 15°; supplement: 105°

57. complement: none because the measure of the angle is greater than 90°; supplement: 60°

58. complement: none because the measure of the angle is greater than 90°; supplement: 20°

59. complement: none because the measure of the angle is greater than 90°; supplement: none because the measure of the angle is greater than 180°

60. complement: none because the measure of the angle is negative; supplement: none because the measure of the angle is negative

In exercises 61−76, use the formulas , ,θ= = ss r v t

21, , and 2

θω ω θ= = =v r A r t

where θ is the

radian measure of the central angle that intercepts an arc of length s in a circle of radius r, v is the linear velocity, ω is the angular velocity, A is the area of a sector of the circle, and t is the time.

61. 7

7 25 0.28 radian 25

s rθ θ θ= ⇒ = ⇒ = =

62. 6

6 5 1.2 radians 5

θ θ θ= ⇒ = ⇒ = =s r

63. 22 10.5 22 44

2.095 radians 10.5 21

s rθ θ

θ

= ⇒ = ⇒

= = ≈

64. 120

120 60 2 radians 60

s rθ θ θ= ⇒ = ⇒ = =

65. First convert the angle measurement from

degrees to radians: 5

25 25 180 36

π π⎛ ⎞° = ° =⎜ ⎟⎝ ⎠° .

Then 5 5

3 1.309 m. 36 12

s r s π πθ= ⇒ = ⋅ = ≈

66. First convert the angle measurement from

degrees to radians: 119

357 357 180 60

π π⎛ ⎞° = ° =⎜ ⎟⎝ ⎠°

Then 119 833

0.7 4.362 m 60 60

s r s π πθ= ⇒ = ⋅ = ≈

67. 6.5 12 78 mθ= ⇒ = ⋅ =s r s

68. 6 3.142 m 6

πθ π= ⇒ = ⋅ = ≈s r s

69. 6 10 60 m/minω= ⇒ = ⋅ =v r v

70. 3.2 5 16 ft/secω= ⇒ = ⋅ =v r v

71. 20 10 2 radians/secω ω ω= ⇒ = ⇒ =v r

72. 5

10 6 radians/min 3

ω ω ω= ⇒ = ⇒ =v r

73. 2 2 2 1 1

10 4 200 in. 2 2

θ= ⇒ ⋅ ⋅ =A r

74. First convert the angle measurement from degrees to radians:

60 60 radians 180 3

π π⎛ ⎞° = ° =⎜ ⎟⎝ ⎠°

( )2 2 21 1 31.5 1.178 ft 2 2 3 8

π πθ ⎛ ⎞= ⇒ = = ≈⎜ ⎟⎝ ⎠A r A

75. First convert the angle measurement from degrees to radians:

60 60 radians 180 3

π π⎛ ⎞° = ° =⎜ ⎟⎝ ⎠°

2 2 21 1 12020 2 2 3 120

6.180 ft

πθ π

π

= ⇒ = ⋅ ⇒ = ⇒

= ≈

A r r r

r

76. 2 2 2 1 1

60 2 60 2 2 60 2 15 7.746 m

θ= ⇒ = ⋅ ⇒ = ⇒

= = ≈

A r r r

r

5.1 Applying the Concepts

77. Three-quarters of a revolution is 3 2π radians. So the arc length (the distance the car moves) is

3 15 70.69 inches

2

πθ ⎛ ⎞= = ≈⎜ ⎟⎝ ⎠s r