478 CHAPTER 7 Analytic Trigonometry However, if the domain of = y x tan is restricted to the interval π π ( ) − 2 , 2 , the restricted function π π = − < < y x x tan 2 2 is one-to-one and so has an inverse function.* See Figure 12. An equation for the inverse of ( ) = = y f x x tan is obtained by interchanging x and y. The implicit form of the inverse function is π π = − < < x y y tan , 2 2 . The explicit form is called the inverse tangent of x and is symbolized by ( ) = = − − y f x x tan 1 1 (or by = y x arctan ). Finding the Exact Value of an Inverse Tangent Function Find the exact value of: (a) − tan 11 (b) ( ) − − tan 3 1 EXAMPLE 6 Figure 12 π π = − < < −∞< <∞ y x x y tan , 2 2 , 2p– 2 2p– 2 p– 2 x y 21 1 x 5 p– 2 x 5 DEFINITION Inverse Tangent Function π π = = −∞< <∞ − < < − y x x y x y tan if and only if tan where and 2 2 1 (3) Here y is the angle whose tangent is x. The domain of the function = − y x tan 1 is −∞< <∞ x , and its range is π π − < < y 2 2 . The graph of = − y x tan 1 can be obtained by reflecting the restricted portion of the graph of = y x tan about the line = y x, as shown in Figure 13(a). Figure 13(b) shows the graphs using a TI-84 Plus CE graphing calculator. Figure 13 y x x y tan , , 2 2 1 π π = −∞< <∞ − < < − (a) –p– 2 –p– 2 p– 2 p– 2 x y –1 1 y = tan–1 x y = tan x y = x 2p– 2 y 5 2p– 2 x 5 p– 2 y 5 p– 2 x 5 (b) 22.5 2.5 2 p 2 p 2 Y2 5 tan(x) Y1 5 tan 21(x) Y3 5 x Now Work PROBLEM 9 6 Find the Value of an Inverse Tangent Function * This is the generally accepted restriction.
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