726 CHAPTER 7 Trigonometric Identities and Equations y 1 (0, 0) –1 –2 y = sin x Restricted domain x –2p 2p –p p – 3p 2 3p 2 –p 2 p 2 [– , ] 2 P 2 P (– , –1) 2 P ( , 1) 2 P Figure 11 Reflecting the graph of y = sin x on the restricted domain, shown in Figure 12(a), across the line y = x gives the graph of the inverse function, shown in Figure 12(b). Some key points are labeled on the graph. The equation of the inverse of y = sin x is found by interchanging x and y to obtain x = sin y. This equation is solved for y by writing y =sin−1 x (read “inverse sine of x”). As Figure 12(b) shows, the domain of y = sin-1 x is 3-1, 14, while the restricted domain of y = sin x, C - p 2 , p 2 D , is the range of y = sin-1 x. An alternative notation for sin−1 x is arcsin x. Restricted domain [– , ] 2 P 2 P 1 –1 p 2 –p 2 (0, 0) y x ( , 1) P 2 ( , –1) P 2 – y = sin x 1 0 y = sin–1 x or y = arcsin x –1 (0, 0) y x (1, ) 2 P (–1, – ) – p 2 p 2 p 3 ( , ) 2 √3 p 4 ( , ) 2 √2 p 4 (– , – ) 2 √2 p 3 (– , – ) 2 √3 p 6 1 2( , ) p 6 (– , – ) 1 2 2 P (b) Figure 12 (a) Inverse Sine Function y =sin−1 x or y =arcsin x means that x = sin y, for - p 2 … y … p 2 . We can think of y =sin−1 x or y =arcsin x as “y is the number (angle) in the interval C−P 2 , P 2 D whose sine is x.” Thus, we can write y = sin-1 x as sin y = x to evaluate it. We must pay close attention to the domain and range intervals.
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