767 CHAPTER 7 Review Exercises Solve each equation for all exact solutions, in radians. 112. sec x 2 = cos x 2 113. cos 2x + cos x = 0 114. 4 sin x cos x = 23 Solve each equation over the interval 30°, 360°2. Write solutions as exact values or to the nearest tenth of a degree, as appropriate. 115. sin2 u + 3 sin u + 2 = 0 116. 2 tan2 u = tan u + 1 117. sin 2u = cos 2u + 1 118. 2 sin 2u = 1 119. 3 cos2 u + 2 cos u - 1 = 0 120. 5 cot2 u - cot u - 2 = 0 Solve each equation for all exact solutions, in degrees. 121. 223 cos u 2 = -3 122. sin u - cos 2u = 0 123. tan u - sec u = 1 Solve each equation for x. 124. 4p - 4 cot-1 x = p 125. 4 3 arctan x 2 = p 126. arccos x = arcsin 2 7 127. arccos x + arctan 1 = 11p 12 128. y = 3 cos x 2 , for x in 30, 2p4 129. y = 1 2 sin x, for x in c - p 2 , p 2 d 130. y = 4 5 sin x - 3 5 , for x in c - p 2 , p 2 d 131. y = 1 2 tan13x + 22, for x in a- 2 3 - p 6 , - 2 3 + p 6b 132. Solve d = 550 + 450 cos a p 50 tb for t, where t is in the interval 30, 504. (Modeling) Solve each problem. 133. Viewing Angle of an Observer A 10-ft-wide chalkboard is situated 5 ft from the left wall of a classroom. See the figure. A student sitting next to the wall x feet from the front of the classroom has a viewing angle of u radians. (a) Show that the value of u is given by y1 = tan-1 a 15 x b - tan-1 a 5 xb . (b) Graph y1 with a graphing calculator to estimate the value of x that maximizes the viewing angle. 134. Snell’s Law Snell’s law states that c1 c2 = sin u1 sin u2 , where c1 is the speed of light in one medium, c2 is the speed of light in a second medium, and u1 and u2 are the angles shown in the figure. Suppose a light is shining up through water into the air as in the figure. As u1 increases, u2 approaches 90°, at which point no light will emerge from the water. Assume the ratio c1 c2 in this case is 0.752. For what value of u1, to the nearest tenth, does u2 = 90°? This value of u1 is the critical angle for water. 5 10 x u Air Water u1 u2
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