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

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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

    0.1995 radians

    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

    Copyright © 2015 Pearson Education Inc.

    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

    Copyright © 2015 Pearson Education Inc.

    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

    Copyright © 2015 Pearson Education Inc.

    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