472 CHAPTER 7 Analytic Trigonometry Grace Hopper (1906–1992) Credit: PJF Military Collection/Alamy Stock Photo Grace Brewster Murray Hopper was a computer pioneer and naval officer. Hopper’s career path was influenced by World War Two. Hopper and her fellow officers in the Harvard lab worked on topsecret calculations essential to the war effort, such as computing rocket trajectories, creating range tables for new anti-aircraft guns, and calibrating minesweepers. Figure 1 y x x y sin , , 1 1 = −∞< <∞ − ≤ ≤ 2p– 2 p– 2 x p 3p –– 2 2p 2p y 21 1 y 5 b, 21 # b # 1 7.1 The Inverse Sine, Cosine, and Tangent Functions • Inverse Functions (Section 5.2, pp. 283–289) • Values of the Trigonometric Functions (Section 6.2, pp. 398–405) • Properties of the Sine, Cosine, and Tangent Functions (Section 6.3, pp. 413–423) • Graphs of the Sine, Cosine, and Tangent Functions (Sections 6.4, pp. 427–436 and 6.5, pp. 443–447) PREPARING FOR THIS SECTION Before getting started, review the following: OBJECTIVES 1 Define the Inverse Sine Function (p. 472) 2 Find the Value of an Inverse Sine Function (p. 474) 3 Define the Inverse Cosine Function (p. 475) 4 Find the Value of an Inverse Cosine Function (p. 476) 5 Define the Inverse Tangent Function (p. 477) 6 Find the Value of an Inverse Tangent Function (p. 478) 7 Use Properties of Inverse Functions to Find Exact Values of Certain Composite Functions (p. 479) 8 Find the Inverse Function of a Trigonometric Function (p. 482) 9 Solve Equations Involving Inverse Trigonometric Functions (p. 482) Now Work the ‘Are You Prepared?’ problems on page 483. In Section 5.2 we discussed inverse functions, and we concluded that if a function is one-to-one, it will have an inverse function. We also observed that if a function is not one-to-one, it may be possible to restrict its domain in some suitable manner so that the restricted function is one-to-one. For example, the function = y x2 is not one-to-one; however, if the domain is restricted to ≥ x 0, the new function is one-to-one. Other properties of a one-to-one function f and its inverse function −f 1 that were discussed in Section 5.2 are summarized next. • ( ) ( ) = −f f x x 1 for every x in the domain of f. • ( ) ( ) = − f f x x 1 for every x in the domain of −f .1 • The domain of = f the range of −f ,1 and the range of = f the domain of −f .1 • The graph of a one-to-one function f and the graph of its inverse −f 1 are symmetric with respect to the line = y x. • If a function ( ) = y f x has an inverse function, the implicit equation of the inverse function is ( ) = x f y . If we solve this equation for y, we obtain the explicit equation ( ) = − y f x . 1 1 Define the Inverse Sine Function Figure 1 shows the graph of = y x sin . Because every horizontal line = y b, where b is between −1 and 1, inclusive, intersects the graph of = y x sin infinitely many times, it follows from the horizontal-line test that the function = y x sin is not one-to-one.
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