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caps lock S D F G shift Z X C fn control option command Section 5.4 Sum and Difference Formulas 403 ns to odd-numbered exercises In Exercises 61-70, prove the identity. 86. tan(x + TT) - cos(x + = 0 61. sin ~ - x) = cos x 62. sin(7 + " + x = COS x 87. sin x + + cos x = 0 63. sin 6 " + x = 2(cos x + 3 sin .x) 88. cos( x - )- sin2 x = 0 64 . cos 4 (ST - x) = -2(cos.x + sin x) 89. HARMONIC MOTION A weight is attached to a 65. cos(T - () + sin(2 1(" + 0 =0 spring suspended vertically from a ceiling. When a driving force is applied to the system, the weight moves 66. tan # - 0 = _1 - tan 0 vertically from its equilibrium position, and this motion is modeled by it value of the expression. 1 + tan 0 67. cos(x + y) cos(x - y) = cos2 x - sin? y y = ~ sin 21 + - cos 21 68. sin(x + y) sin(x - y) = sin? x = sin? y 3 7 69. sin(x + y) + sin(x - y) = 2 sin x cos y where y is the distance from equilibrium (in feet) and 16 70. cos(x + y) + cos(x - y) = 2 cos x cos y is the time (in seconds). (a) Use the identity sin 600 In Exercises 71-74, simplify the expression algebraically and sin 30 use a graphing utility to confirm your answer graphically. a sin BO + b cos BO = Va2 + b2 sin(Be + C) where C = arctan(b/a), a > 0, to write the mode 71. cos 2 - x 72. COS( TI + x) in the form y = vaz + be sin(Bi + C). (b) Find the amplitude of the oscillations of the weigh 73. sin( 2 74. tan( 7 + 0) (c) Find the frequency of the oscillations of the weigh 90. STANDING WAVES The equation of a standing wa In Exercises 75-84, find all solutions of the equation in the is obtained by adding the displacements of two wav act value of the trigonometric interval [0, 2 TT). traveling in opposite directions (see figure). Assur and cos v = -5. (Both u andy that each of the waves has amplitude A, period T, 75. sin(x + 7) - sin x + 1 =0 wavelength A. If the models for these waves are 44. cos(u 76. sin(x + 7) - sin x - 1 = 0 17. cos(x + 17) - cos x - 1 = 0 y1 = A cos 27 7 - and y2 = A cos 27 ( 46. sin(v - u) 78. cos(x + 17) - cos x + 1 = 0 48. csc(u - v) show that 50. cot(u + v) 79. sin x + - sin( x - #) = y1 + y2 = 2A cos 2TIX T - cos