SECTION 7.1 The Inverse Sine, Cosine, and Tangent Functions 473 However, if the domain of = y x sin is restricted to the interval π π ⎡ − ⎣ ⎢ ⎤ ⎦ ⎥ 2 , 2 , the restricted function π π = − ≤ ≤ y x x sin 2 2 is one-to-one and has an inverse function.* See Figure 2. An equation for the inverse of ( ) = = y f x x sin is obtained by interchanging x and y. The implicit form of the inverse function is π π = − ≤ ≤ x y y sin , 2 2 . The explicit form is called the inverse sine of x and is symbolized by ( ) = = − − y f x x sin . 1 1 Figure 2 y x x y sin , 2 2 , 1 1 π π = − ≤ ≤ − ≤ ≤ 2p– 2 2p– 2 p– 2 p– 2 p– 2 x 21 1 y 21 1 , 21 ( ) ( ) p– 2 , 1 2 Because the restricted domain of the sine function is π π ⎡ − ⎣ ⎢ ⎤ ⎦ ⎥ 2 , 2 , the range of the inverse sine function is π π ⎡ − ⎣ ⎢ ⎤ ⎦ ⎥ 2 , 2 , and because the range of the sine function is [ ] −1, 1 , the domain of the inverse sine function is [ ] −1, 1 . DEFINITION Inverse Sine Function π π = = − ≤ ≤ − ≤ ≤ − y x x y x y sin ifandonlyif sin where 1 1 and 2 2 1 (1) The inverse sine of = − x y x , sin , 1 is read as “ y is the angle or real number whose sine equals x,” or, alternatively, is read “ y is the inverse sine of x.” Be careful about the notation used.The superscript −1 that appears in = − y x sin 1 is not an exponent but the symbol used to denote the inverse function −f 1 of f. (To avoid confusion, some texts use the notation = y x arcsin instead of = − y x sin . 1 ) The graph of the inverse sine function can be obtained by reflecting the restricted portion of the graph of ( ) = = y f x x sin about the line = y x, as shown in Figure 3(a). Figure 3(b) shows the graph using a TI-84 Plus CE graphing calculator. Figure 3 y x x y sin , 1 1, 2 2 1 π π = − ≤ ≤ − ≤ ≤ − (b) 2 p 2 p 2 Y1 5 sin 21x Y2 5 sinx Y3 5 x (1, ) (–1, ) ( , –1) ( , 1) –p– 2 –p– 2 –p– 2 p– 2 p– 2 x –1 1 y –1 1 –p– 2 p– 2 p– 2 y = sin x y = x y = sin–1x (a) 22.5 2.5 Now Work PROBLEM 5 *Although there are many other ways to restrict the domain and obtain a one-to-one function, mathematicians have agreed to use the interval π π ⎡ − ⎣ ⎢ ⎤ ⎦ ⎥ 2 , 2 to define the inverse of = y x sin .
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