738 CHAPTER 7 Trigonometric Identities and Equations (e) Graph the function for u with a graphing calculator, and determine the distance that maximizes the angle. (f ) The concept in part (e) was first investigated in 1471 by the astronomer Regiomontanus. (Data from Maor, E., Trigonometric Delights, Princeton University Press.) If the bottom of the picture is a meters above eye level and the top of the picture is b meters above eye level, then the optimum value of x is 2ab meters. Use this result to find the exact answer to part (e). 108. Landscaping Formula A shrub is planted in a 100-ft-wide space between buildings measuring 75 ft and 150 ft tall. The location of the shrub determines how much sun it receives each day. Show that if u is the angle in the figure and x is the distance of the shrub from the taller building, then the value of u (in radians) is given by u = p - arctan a 75 100 - xb - arctan a 150 x b . Write each trigonometric expression as an algebraic expression in u, for u 70. See Example 7. 95. sin1arccos u2 96. tan1arccos u2 97. cos1arcsin u2 98. cot1arcsin u2 99. sin a2 sec-1 u 2b 100. cos a2 tan-1 3 ub 101. tan asin-1 u 2u2 + 2 b 102. sec acos-1 u 2u2 + 5b 103. sec ¢arccot 24 - u2 u ≤ 104. csc ¢arctan 29 - u2 u ≤ (Modeling) Solve each problem. 105. Angle of Elevation of a Shot Put Refer to Example 8. Suppose a shot-putter can consistently release the steel ball with velocity v of 32 ft per sec from an initial height h of 5.0 ft. What angle, to the nearest degree, will maximize the distance? 106. Angle of Elevation of a Shot Put Refer to Example 8. (a) What is the optimal angle, to the nearest degree, when h = 0? (b) Fix h at 6 ft and regard u as a function of v. As v increases without bound, the graph approaches an asymptote. Find the equation of that asymptote. 107. Observation of a Painting A painting 1 m high and 3 m from the floor will cut off an angle u to an observer, where u = tan-1 a x x2 + 2b , assuming that the observer is x meters from the wall where the painting is displayed and that the eyes of the observer are 2 m above the ground. (See the figure.) Find the value of u for each value of x. Round to the nearest degree. (a) 1 (b) 2 (c) 3 (d) Derive the formula given above. (Hint: Use the identity for tan1u + a2. Use right triangles.) 1 1 3 2 x a U 100 ft 75 ft 150 ft x u
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