750 CHAPTER 7 Trigonometric Identities and Equations 81. 8 sec2 x 2 = 4 82. sin2 x 2 - 2 = 0 83. sin u 2 = csc u 2 84. sec u 2 = cos u 2 85. cos 2x + cos x = 0 86. sin x cos x = 1 4 Solve each equation (x in radians and u in degrees) for all exact solutions where appropriate. Round approximate answers in radians to four decimal places and approximate answers in degrees to the nearest tenth. Write answers using the least possible nonnegative angle measures. See Examples 6–8. 87. 22 sin 3x - 1 = 0 88. -2 cos 2x = 23 89. cos u 2 = 1 90. sin u 2 = 1 91. 223 sin x 2 = 3 92. 223 cos x 2 = -3 93. 2 sin u = 2 cos 2u 94. cos u - 1 = cos 2u 95. 1 - sin x = cos 2x 96. sin 2x = 2 cos2 x 97. 3 csc2 x 2 = 2 sec x 98. cos x = sin2 x 2 99. 2 - sin 2u = 4 sin 2u 100. 4 cos 2u = 8 sin u cos u 101. 2 cos2 2u = 1 - cos 2u 102. sin u - sin 2u = 0 The following equations cannot be solved by algebraic methods. Use a graphing calculator to find all solutions over the interval 30, 2p2. Express solutions to four decimal places. 103. 2 sin 2x - x3 + 1 = 0 104. 3 cos x 2 + 2x - 2 = - 1 2 x + 2 (Modeling) Solve each problem See Examples 9 and 10. 105. Pressure on the Eardrum A pure tone has a unique constant frequency and amplitude that sounds rather dull. Pressures caused by pure tones on the eardrum are sinusoidal. The change in pressure P in pounds per square foot on a person’s eardrum from the pure tone middle C at time t in seconds can be modeled using the equation P = 0.004 sin c 2p1261.632t + p 7 d . When P is positive, there is an increase in pressure and the eardrum is pushed inward. When P is negative, there is a decrease in pressure and the eardrum is pushed outward. (Data from Roederer, J., Introduction to the Physics and Psychophysics of Music, Second Edition, Springer-Verlag.) (a) Determine algebraically the values of t for which P = 0 over 30, 0.0054. (b) From a graph and the answer in part (a), determine the interval for which P … 0 over 30, 0.0054. (c) Would an eardrum hearing this tone be vibrating outward or inward when P60? 106. Accident Reconstruction To reconstruct accidents in which a vehicle vaults into the air after hitting an obstruction, the model 0.342D cos u + h cos2 u = 16D2 V0 2 can be used. V0 is velocity in feet per second of the vehicle when it hits the obstruction, D is distance (in feet) from the obstruction to the landing point, and h is the difference in height (in feet) between landing point and takeoff point. Angle u is the takeoff angle, the angle between the horizontal and the path of the vehicle. Find u to the nearest degree if V0 = 60, D= 80, and h = 2.
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