476 CHAPTER 7 Analytic Trigonometry An equation for the inverse of ( ) = = y f x x cos is obtained by interchanging x and y.The implicit form of the inverse function is π = ≤ ≤ x y y cos , 0 . The explicit form is called the inverse cosine of x and is symbolized by ( ) = = − − y f x x cos 1 1 (or by = y x arccos ). Find the Exact Value of an Inverse Cosine Function Find the exact value of: − cos 01 Solution EXAMPLE 4 Let θ = − cos 0. 1 Then θ is the angle, θ π ≤ ≤ 0 , whose cosine equals 0. θ θ π θ θ π = ≤ ≤ = ≤ ≤ − cos 0 0 cos 0 0 1 DEFINITION Inverse Cosine Function π = = − ≤ ≤ ≤ ≤ − y x x y x y cos if and only if cos where 1 1 and 0 1 (2) Here y is the angle whose cosine is x. Because the range of the cosine function, = y x cos , is − ≤ ≤ y 1 1, the domain of the inverse function = − y x cos 1 is − ≤ ≤ x 1 1. Because the restricted domain of the cosine function, = y x cos , is π ≤ ≤ x 0 , the range of the inverse function = − y x cos 1 is π ≤ ≤ y 0 . The graph of = − y x cos 1 can be obtained by reflecting the restricted portion of the graph of = y x cos about the line = y x, as shown in Figure 8(a) . Figure 8(b) shows the graphs using Desmos. Figure 8 y x x y cos , 1 1, 0 1 π = − ≤ ≤ ≤ ≤ − p– 2 p– 2 x p (1, 0) (a) (–1, p) –1 y –1 p (0, 1) (p, –1) y = cos–1 x y = cos x y = x (b) Now Work PROBLEM 7 4 Find the Value of an Inverse Cosine Function
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