735 7.5 Inverse Circular Functions EXAMPLE 8 Finding Optimal Angle of Elevation of a Shot Put The optimal angle of elevation u for a shot-putter to achieve the greatest distance depends on the velocity v of the throw and the initial height h of the shot. See Figure 28. One model for u that attains this greatest distance is u = arcsin aA v2 2v2 + 64h b . (Data from Townend, M. S., Mathematics in Sport, Chichester, Ellis Horwood Ltd.) h D u Figure 28 An athlete can consistently put the shot with h = 6.6 ft and v = 42 ft per sec. At what angle should the athlete release the ball to maximize distance? SOLUTION To find this angle, substitute and use a calculator in degree mode. u = arcsin aA 422 214222 + 6416.62 b ≈42° Use h = 6.6, v = 42, and a calculator. S Now Try Exercise 105. 7.5 Exercises CONCEPT PREVIEW Fill in the blank(s) to correctly complete each sentence. 1. For a function to have an inverse, it must be -to- . 2. The domain of y = arcsin x equals the of y = sin x. 3. y = cos-1 x means that x = for 0 … y … p. 4. The point Ap 4 , 1B lies on the graph of y = tan x. Therefore, the point lies on the graph of y = tan-1 x. 5. If a function f has an inverse and ƒ1p2 = -1, then ƒ-11-12 = . 6. To evaluate sec-1 x, use the value of cos-1 . CONCEPT PREVIEW Write a short answer for each of the following. 7. Consider the inverse sine function y = sin-1 x, or y = arcsin x. (a) What is its domain? (b) What is its range? (c) Is this function increasing or decreasing? (d) Why is arcsin1-22 not defined? 1
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