723 7.4 Double-Angle and Half-Angle Identities 103. sin x 1 + cos x 104. 1 - cos x sin x 105. tan x 2 + cot x 2 cot x 2 - tan x 2 106. 1 - 8 sin2 x 2 cos2 x 2 (Modeling) Solve each problem. See Example 5. 107. Wattage Consumption Use the identity cos 2u = 1 - 2 sin2 u to determine values of a, c, and v so that the equation W= 1163 sin 120pt22 15 becomes W= a cos1vt2 + c. Round to the nearest tenth as necessary. Check by graphing both expressions for W on the same coordinate axes. 108. Amperage, Wattage, and Voltage Amperage is a measure of the amount of electricity that is moving through a circuit, and voltage is a measure of the force pushing the electricity. The wattage W consumed by an electrical device can be determined by calculating the product of the amperage I and voltage V. (Data from Wilcox, G., and C. Hesselberth, Electricity for Engineering Technology, Allyn & Bacon.) (a) A household circuit has voltage V = 163 sin 120pt when an incandescent light bulb is turned on with amperage I = 1.23 sin 120pt. Graph the wattage W= VI consumed by the light bulb in the window 30, 0.054 by 3-50, 3004. (b) Determine the maximum and minimum wattages used by the light bulb. (c) Use identities to determine values for a, c, and v so that W= a cos1vt2 + c. (d) Check by graphing both expressions for W on the same coordinate axes. (e) Use the graph to estimate the average wattage used by the light. For how many watts (to the nearest integer) would this incandescent light bulb be rated? (Modeling) Mach Number An airplane flying faster than the speed of sound sends out sound waves that form a cone, as shown in the figure. The cone intersects the ground to form a hyperbola. As this hyperbola passes over a particular point on the ground, a sonic boom is heard at that point. If u is the angle at the vertex of the cone, then sin u 2 = 1 m , where m is the Mach number for the speed of the plane. (We assume m71.) The Mach number is the ratio of the speed of the plane to the speed of sound. Thus, a speed of Mach 1.4 means that the plane is flying at 1.4 times the speed of sound. In each exercise, u or m is given. Find the other value (u to the nearest degree and m to the nearest tenth, as applicable). 109. m= 5 4 110. m= 3 2 111. u = 60° 112. u = 30° u
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