42. Think About a Plan The function h = 25 sin ( π 20 ( t − 10 ) ) + 34 h equals 25 sine . open . pi over 20 . open , t minus 10 , close . close . plus 34 models the height h of a Ferris wheel car in feet, t seconds after starting. When will the car first be 30 ft off the ground?
    • What is the inverse function?
    • How can the inverse function help you answer the question?
43. Electricity The function I = 40 sin 60 π t i equals 40 sine , 60 pi t models the current I in amps that an electric generator is producing after t seconds. When is the first time that the current will reach 20 amps? − 20 negative 20 amps?

44. Reasoning The graphing calculator screen shows a portion of the graphs of y = sin θ y equals sine theta and y = 0.5 . y equals 0.5 .

    
    Image Long Description

    1. Write the complete solution of sin θ ≥ 0.5 . sine theta greater than or equal to 0.5 .
    2. Write the complete solution of sin θ ≤ 0 . 5 . sine theta less than or equal to 0 . 5 .
    3. Writing Explain how you can solve inequalities involving trigonometric functions.

Find the complete solution in radians of each equation.

45. 2 sin 2 θ + cos θ − 1 = 0 2 , sine squared , theta plus cosine theta negative 1 equals 0
46. sin 2 θ − 1 = cos 2 θ sine squared , theta negative 1 equals , cosine squared , theta
47. 2 sin θ + 1 = csc θ 2 sine theta plus 1 equals co-secant theta
48. 3 tan 2 θ − 1 = sec 2 θ 3 , tangent squared , theta negative 1 equals , secant squared , theta
49. sin θ cos θ = 1 2 cos θ sine theta cosine theta equals , 1 half cosine theta
50. tan θ sin θ = 3 sin θ tangent theta sine theta equals 3 sine theta
51. 2 cos 2 θ + sin θ = 1 2 , cosine squared , theta plus sine theta equals 1
52. sin θ cot 2 θ − 3 sin θ = 0 sine theta , co-tangent squared , theta negative 3 sine theta equals 0
53. 4 sin 2 θ + 1 = 4 sin θ 4 , sine squared , theta plus 1 equals 4 sine theta

Find the x-intercepts of the graph of each function.

54. y = 2 cos θ + 1 y equals 2 cosine theta plus 1
55. y = 2 sin 2 θ − 1 y equals 2 , sine squared , theta negative 1
56. y = cos 2 θ − 1 y equals , cosine squared , theta negative 1
57. y = tan 2 θ − 1 y equals , tangent squared , theta negative 1
58. y = 2 sin 4 θ − sin 2 θ y equals 2 , sine to the fourth , theta negative , sine squared , theta
59. y = 2 cos 2 θ − 3 cos θ − 2 y equals 2 , cosine squared , theta negative 3 cosine theta negative 2
60. Find the complete solution of sin 2 θ + 2 sin θ + 1 = 0 . sine squared , theta plus 2 sine theta plus 1 equals 0 . (Hint: How would you solve x 2 + 2 x + 1 = 0 ? ) x squared , plus 2 x plus 1 equals 0 question mark close
61. 1. Open-Ended Write three trigonometric equations each with the complete solution π + 2 π n . pi plus 2 pi n .
    2. Describe how you found the equations in part (a).

C Challenge

Solve each trigonometric equation for θ theta in terms of y.

Sample y = 2 sin 3 θ + 4 y equals 2 sine 3 theta plus 4

y = 2 sin 3 θ + 4 sin 3 θ = y − 4 2 3 θ = sin − 1 ( y − 4 2 ) + 2 π n , 2 ≤ y ≤ 6 θ = 1 3 sin − 1 ( y − 4 2 ) + 2 π 3 n , 2 ≤ y ≤ 6 table with 4 rows and 2 columns , row1 column 1 , y , column 2 equals 2 sine 3 theta plus 4 , row2 column 1 , sine 3 theta , column 2 equals . fraction y minus 4 , over 2 end fraction , row3 column 1 , 3 theta , column 2 equals . sine super negative 1 end super . open . fraction y minus 4 , over 2 end fraction . close . plus 2 pi n comma 2 less than or equal to y less than or equal to 6 , row4 column 1 , theta , column 2 equals , 1 third . sine super negative 1 end super . open . fraction y minus 4 , over 2 end fraction . close . plus , fraction 2 pi , over 3 end fraction , n comma 2 less than or equal to y less than or equal to 6 , end table

62. y = cos 2 θ y equals cosine 2 theta
63. y = 3 sin ( θ + 2 ) y equals 3 sine open theta plus 2 close
64. y = − 4 cos 2 π θ y equals negative 4 cosine 2 pi theta
65. y = 2 cos θ + 1 y equals 2 cosine theta plus 1

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