728 CHAPTER 7 Trigonometric Identities and Equations Figure 15 y x (0, 1) ( , 0) P 2 (P, –1) Restricted domain [0, P] 1 –1 0 –p 2 p 2 p y = cos x Figure 16 y x 1 0 y = cos–1 x or y = arccos x (1, 0) –1 (–1, P) p p 6 ( , ) 2 √3 p 4 ( , ) 2 √2 p 3 1 2( , ) 2p 3 (– , ) 2 √2 3p 4 (– , ) 2 √3 5p 6 1 2 (– , ) (0, ) P 2 Inverse Cosine Function y =cos−1 x or y =arccos x means that x = cos y, for 0 … y … p. We can think of y =cos−1 x or y =arccos x as “y is the number (angle) in the interval 3 0, P4 whose cosine is x.” EXAMPLE 2 Finding Inverse Cosine Values Find the value of each real number y if it exists. (a) y = arccos 1 (b) y = cos-1 ¢- 22 2 ≤ SOLUTION (a) Because the point 11, 02 lies on the graph of y = arccos x in Figure 16, the value of y, or arccos 1, is 0. Alternatively, we can think of y = arccos 1 as “y is the number in 30, p4 whose cosine is 1,” or cos y = 1. Thus y = 0, since cos 0 = 1 and 0 is in the range of the arccosine function. (b) We must find the value of y that satisfies cos y = - 22 2 , where y is in the interval 30, p4, which is the range of the function y = cos-1 x. The only value for y that satisfies these conditions is 3p 4 . Again, this can be verified from the graph in Figure 16. S Now Try Exercises 15 and 23. −1 1 0 p These screens support the results of Example 2 because - 22 2 ≈ -0.7071068 and 3p 4 ≈2.3561945. −1 1 0 p Inverse Cosine Function The function y =cos−1 x or y =arccos x is defined by restricting the domain of the function y = cos x to the interval 30, p4 as in Figure 15. This restricted function, which is the part of the graph in Figure 15 shown in color, is one-to-one and has an inverse function. The inverse function, y = cos-1 x, is found by interchanging the roles of x and y. Reflecting the graph of y = cos x across the line y = x gives the graph of the inverse function shown in Figure 16. Some key points are shown on the graph.
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