674 CHAPTER 6 The Circular Functions and Their Graphs 58. (Modeling) Phase Angle of the Moon Because the moon orbits Earth, we observe different phases of the moon during the period of a month. In the figure, t is the phase angle. Earth Sun Moon t The phase F of the moon is modeled by F1t2 = 1 2 11 - cos t2 and gives the fraction of the moon’s face that is illuminated by the sun. (Data from Duffet-Smith, P., Practical Astronomy with Your Calculator, Cambridge University Press.) Evaluate each expression and interpret the result. (a) F102 (b) F a p 2b (c) F1p2 (d) F a 3p 2 b 59. Concept Check Which one of the following statements is true about the graph of y = 4 sin 2x? A. It has amplitude 2 and period p 2 . B. It has amplitude 4 and period p. C. Its range is 30, 44. D. Its range is 3-4, 04. 60. Concept Check Which one of the following statements is false about the graph of y = -3 cos 1 2 x? A. Its range is 3-3, 34. B. Its domain is 1-∞, ∞2. C. It has amplitude 3 and period 4p. D. It has amplitude -3 and period p. For each function, give the amplitude, period, vertical translation, and phase shift, as applicable. 61. y = 2 sin x 62. y = tan 3x 63. y = - 1 2 cos 3x 64. y = 2 sin 5x 65. y = 1 + 2 sin 1 4 x 66. y = 3 - 1 4 cos 2 3 x 67. y = 3 cos ax + p 2b 68. y = -sin ax - 3p 4 b 69. y = 1 2 csc a2x - p 4b 70. y = 2 sec1px - 2p2 71. y = 1 3 tan a3x - p 3b 72. y = cot a x 2 + 3p 4 b Concept Check Identify the circular function that satisfies each description. 73. period is p; x-intercepts have x-values of the form np, where n is any integer 74. period is 2p; graph passes through the origin 75. period is 2p; graph passes through the point A p 2 , 0B 76. period is 2p; domain is 5x x ≠np, where n is any integer6 77. period is p; function is decreasing on the interval 10, p2 78. period is 2p; has vertical asymptotes of the form x = 12n + 12 p 2 , where n is any integer
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