729 7.5 Inverse Circular Functions Inverse Cosine Function y =cos −1x or y =arccos x Domain: 3-1, 14 Range: 30, p4 • The inverse cosine function is decreasing on the open interval 1-1, 12 and continuous on its domain 3-1, 14. • Its x-intercept is 11, 02 and its y-intercept is A0, p 2 B. • Its graph is not symmetric with respect to either the y-axis or the origin. x y -1 p - 12 2 3p 4 0 p 2 1 2 2 p 4 1 0 Figure 17 x y y = cos–1 x –1 0 1 p −1 1 0 p y = cos–1x Inverse Tangent Function Restricting the domain of the function y = tan x to the open interval A - p 2 , p 2 B yields a one-to-one function. By interchanging the roles of x and y, we obtain the inverse tangent function given by y =tan−1 x or y =arctan x. Figure 18 shows the graph of the restricted tangent function. Figure 19 gives the graph of y = tan-1 x. Figure 18 y x 0 y = tan x (0, 0) Restricted domain ( , 1) 4 P (– , –1) 4 P –p 2 p 2 (– , ) 2 P 2 P –1 1 –2 2 Figure 19 y x –1 (0, 0) y = tan–1 x or y = arctan x 1 2 –2 p 6 ( , ) 3 √3 (1, ) (–1, – ) p 3 (–√3 , – ) p 6 (– , – ) 3 √3 – p 2 p 4 p 2 –p 4 4 P 4 P p 3 (√3 , ) InverseTangent Function y =tan−1 x or y =arctan x means that x = tan y, for - p 2 6y 6 p 2 . We can think of y =tan−1 x or y =arctan x as “y is the number (angle) in the interval A−P 2 , P 2 B whose tangent is x.”
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