730 CHAPTER 7 Trigonometric Identities and Equations We summarize this discussion about the inverse tangent function as follows. Other Inverse Circular Functions The other three inverse trigonometric functions are defined similarly. Their graphs are shown in Figure 21. Inverse Cotangent, Secant, and Cosecant Functions* y =cot−1 x or y =arccot x means that x = cot y, for 0 6y 6p. y =sec−1 x or y =arcsec x means that x = sec y, for 0 … y … p, y ≠ p 2 . y =csc−1 x or y =arccsc x means that x = csc y, for - p 2 … y … p 2 , y ≠0. InverseTangent Function y =tan−1 x or y =arctan x Domain: 1-∞, ∞2 Range: A - p 2 , p 2 B • The inverse tangent function is increasing on 1-∞, ∞2 and continuous on its domain 1-∞, ∞2. • 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, tan-11-x2 = -tan-1 x. • The lines y = p 2 and y = - p 2 are horizontal asymptotes. x y -1 - p 4 - 13 3 - p 6 0 0 1 3 3 p 6 1 p 4 Figure 20 x y y = tan–1 x –2 –1 0 1 2 – p 2 p 2 −4 4 p 2 − p 2 y = tan–1x –2 2 –1 0 1 y y = cot–1 x p x –2 2 –1 0 1 y y = sec–1 x p x y x y = csc–1 x –2 –1 0 p 2 1 2 –p 2 Figure 21 * The inverse secant and inverse cosecant functions are sometimes defined with different ranges. We use intervals that match those of the inverse cosine and inverse sine functions, respectively (except for one missing point).
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