727 7.5 Inverse Circular Functions LOOKING AHEAD TO CALCULUS The inverse circular functions are used in calculus to solve certain types of related-rates problems and to integrate certain rational functions. EXAMPLE 1 Finding Inverse Sine Values Find the value of each real number y if it exists. (a) y = arcsin 1 2 (b) y = sin-11-12 (c) y = sin-11-22 ALGEBRAIC SOLUTION (a) The graph of y = arcsin x (Figure 12(b)) includes the point A1 2 , p 6 B . Therefore, arcsin 1 2 = p 6. Alternatively, we can think of y = arcsin 1 2 as “y is the number in C - p 2 , p 2 D whose sine is 1 2 .” Then we can write the given equation as sin y = 1 2 . Because sin p 6 = 1 2 and p 6 is in the range of the arcsine function, y = p 6 . (b) Writing the equation y = sin-11-12 in the form sin y = -1 shows that y = - p 2 . Notice that the point A -1, - p 2 B is on the graph of y = sin-1 x. (c) Because -2 is not in the domain of the inverse sine function, sin-11-22 does not exist. GRAPHING CALCULATOR SOLUTION (a)–(c) We graph the equation y1 = sin-1 x and find the points with x-values 1 2 = 0.5 and -1. For these two x-values, Figure 13 indicates that y = p 6 ≈0.52359878 and y = - p 2 ≈ -1.570796. Because sin-11-22 does not exist, a calculator will give an error message for this input. S Now Try Exercises 13, 21, and 25. −1 1 p 2 − p 2 −1 1 p 2 − p 2 Figure 13 CAUTION In Example 1(b), sin 3p 2 = -1, but 3p 2 is not in the range of the inverse sine function. Be certain that the number given for an inverse function value is in the range of the inverse function being considered. Inverse Sine Function y =sin−1 x or y =arcsin x Domain: 3-1, 14 Range: C - p 2 , p 2 D x y y = sin–1 x –1 0 1 – p 2 p 2 −1 1 p 2 − p 2 y = sin–1x x y -1 - p 2 - 12 2 - p 4 0 0 1 2 2 p 4 1 p 2 Figure 14 • The inverse sine function is increasing on the open interval 1-1, 12 and continuous on its domain 3-1, 14. • Its x- and y-intercepts are both 10, 02. • Its graph is symmetric with respect to the origin, so the function is an odd function. For all x in the domain, sin-11-x2 = -sin-1 x.
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