759 7.7 Equations Involving Inverse Trigonometric Functions 56. Programming Language for Inverse Functions In some programming languages, the only inverse trigonometric function available is arctangent. The other inverse trigonometric functions can be expressed in terms of arctangent. (a) Let u = arcsin x. Solve the equation for x in terms of u. (b) Use the result of part (a) to label the three sides of the triangle in the figure in terms of x. (c) Use the triangle from part (b) to write an equation for tan u in terms of x. (d) Solve the equation from part (c) for u. 57. Alternating Electric Current In the study of alternating electric current, instantaneous voltage is modeled by the equation E = Emax sin 2pƒt, where f is the number of cycles per second, Emax is the maximum voltage, and t is time in seconds. (a) Solve the equation for t. (b) Find the least positive value of t if Emax = 12, E = 5, and f = 100. Use a calculator and round to two significant digits. 58. Viewing Angle of an Observer While visiting a museum, an observer views a painting that is 3 ft high and hangs 6 ft above the ground. See the figure. Assume her eyes are 5 ft above the ground, and let x be the distance in feet from the spot where she is standing to the wall displaying the painting. x ft 5 ft u 6 ft 3 ft (a) Show that u, the viewing angle subtended by the painting, is given by the equation u = tan-1 a 4 xb - tan-1 a 1 xb . (b) Find the value of x to the nearest hundredth for each value of u. (i) u = p 6 (ii) u = p 8 (c) Find the value of u to the nearest hundredth for each value of x. (i) x = 4 (ii) x = 3 59. Movement of an Arm In the following equation, t is time (in seconds) and y is the angle formed by a rhythmically moving arm. y = 1 3 sin 4pt 3 (a) Solve the equation for t. (b) At what time, to the nearest hundredth of a second, does the arm first form an angle of 0.3 radian? 60. Provide graphical support for the solution in Example 4 by showing that the graph of y = sin-1 x - cos-1 x - p 6 has a zero of 23 2 ≈0.8660254. X Y 0 u
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