484 CHAPTER 7 Analytic Trigonometry 47. π ( ) − tan tan 4 5 1 48. π ( ) ⎡ − ⎣ ⎢ ⎤ ⎦ ⎥ − tan tan 10 9 1 49. π ( ) ⎡ − ⎣ ⎢ ⎤ ⎦ ⎥ − tan tan 2 3 1 50. π ( ) − cos cos 4 3 1 51. π ( ) ⎡ − ⎣ ⎢ ⎤ ⎦ ⎥ − cos cos 4 1 52. π ( ) ⎡ − ⎣ ⎢ ⎤ ⎦ ⎥ − sin sin 3 4 1 53. π( ) ⎡ ⎣ ⎢ ⎤ ⎦ ⎥ − tan tan 2 1 54. π ( ) ⎡ − ⎣ ⎢ ⎤ ⎦ ⎥ − tan tan 3 2 1 55. ( ) − sin sin 1 4 1 56. ( ) ⎡ − ⎣ ⎢ ⎤ ⎦ ⎥ − cos cos 2 3 1 57. ( ) − tan tan 41 58. [ ] ( ) − − tan tan 2 1 59. ( ) − cos cos 1.2 1 60. [ ] ( ) − − sin sin 2 1 61. π ( ) − tan tan 1 62. [ ] ( ) − − sin sin 1.5 1 In Problems 63–70, find the inverse function −f 1 of each function f. Find the range of f and the domain and range of −f .1 63. ( ) = + f x x 5 sin 2; π π − ≤ ≤ x 2 2 64. ( ) = − f x x 2 tan 3; π π − < < x 2 2 65. ( ) ( ) = − f x x 2 cos 3 ; π ≤ ≤ x 0 3 66. π π ( ) ( ) = − ≤ ≤ f x x x 3 sin 2 ; 4 4 67. π π ( ) ( ) = − + − − − < < − f x x x tan 1 3; 1 2 2 1 68. π ( ) ( ) = + + − ≤ ≤ − f x x x cos 2 1; 2 2 69. π π ( ) ( ) = + − − ≤ ≤− + f x x x 3 sin 2 1 ; 1 2 4 1 2 4 70. π ( ) ( ) = + − ≤ ≤− + f x x x 2 cos 3 2 ; 2 3 2 3 3 In Problems 71–78, find the exact solution of each equation. 71. π = − x 4 sin 1 72. π = − x 2 cos 1 73. π ( ) = − x 3 cos 2 2 1 74. π ( ) − = − x 6 sin 3 1 75. π = − x 3 tan 1 76. π − = − x 4 tan 1 77. π − = − − x x 4 cos 2 2 cos 1 1 78. π π − = − − − x x 5 sin 2 2 sin 3 1 1 Applications and Extensions In Problems 79–84, use the following discussion. The formula θ π ( ) = − ⎡ ⎣ ⎢ ⎤ ⎦ ⎥ − D i 24 1 cos tan tan 1 can be used to approximate the number of hours of daylight D when the declination of the Sun is °i at a location θ° north latitude for any date between the vernal equinox and autumnal equinox. The declination of the Sun is defined as the angle i between the equatorial plane and any ray of light from the Sun. The latitude of a location is the angle θ between the Equator and the location on the surface of Earth, with the vertex of the angle located at the center of Earth. See the figure.To use the formula, θ ( ) − i cos tan tan 1 must be expressed in radians. (b) Vernal equinox ( ) = ° i 0 (c) July 4 ( ) = ° ′ i 22 48 80. Approximate the number of hours of daylight in New York, New York ( ° ′ 40 45 north latitude), for the following dates: (a) Summer solstice ( ) = ° i 23.5 (b) Vernal equinox ( ) = ° i 0 (c) July 4 ( ) = ° ′ i 22 48 81. Approximate the number of hours of daylight in Honolulu, Hawaii ( ° ′ 21 18 north latitude), for the following dates: (a) Summer solstice ( ) = ° i 23.5 (b) Vernal equinox ( ) = ° i 0 (c) July 4 ( ) = ° ′ i 22 48 82. Approximate the number of hours of daylight in Anchorage, Alaska ( ° ′ 61 10 north latitude), for the following dates: (a) Summer solstice ( ) = ° i 23.5 (b) Vernal equinox ( ) = ° i 0 (c) July 4 ( ) = ° ′ i 22 48 83. Approximate the number of hours of daylight at the Equator ( °0 north latitude) for the following dates: (a) Summer solstice ( ) = ° i 23.5 (b) Vernal equinox ( ) = ° i 0 (c) July 4 ( ) = ° ′ i 22 48 (d) What do you conclude about the number of hours of daylight throughout the year for a location at the Equator? 79. Approximate the number of hours of daylight in Houston, Texas ( ° ′ 29 45 north latitude), for the following dates: (a) Summer solstice ( ) = ° i 23.5 N Pole Equator i 8 N Pole Equator u8 Sun u8 North latitude
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