482 CHAPTER 7 Analytic Trigonometry 8 Find the Inverse Function of a Trigonometric Function NOTE The range of f also can be found using transformations. The range of y x sin = is 1, 1 . [ ] − The range of = y x 2sin is 2, 2 [ ] − due to the vertical stretch by a factor of 2. The range of ( ) = − f x x 2sin 1 is [ ] −3, 1 due to the shift down of 1 unit. j Finding the Inverse Function of a Trigonometric Function (a) Find the inverse function π π ( ) = − − ≤ ≤ −f f x x x of 2 sin 1, 2 2 . 1 (b) Find the range of f and the domain and range of −f .1 Solution EXAMPLE 10 (a) The function f is one-to-one and so has an inverse function. Follow the steps on page 288 for finding the inverse function. = − = − + = = + = + − y x x y x y y x y x 2 sin 1 2 sin 1 1 2 sin sin 1 2 sin 1 2 1 The inverse function is ( ) = + − − f x x sin 1 2 . 1 1 (b) To find the range of f, use the fact that the domain of −f 1 equals the range of f. Since the domain of the inverse sine function is the interval [ ] −1, 1 , the argument +x 1 2 must be in the interval [ ] −1, 1 . − ≤ + ≤ − ≤ + ≤ − ≤ ≤ x x x 1 1 2 1 2 1 2 3 1 The domain of −f 1 is { } − ≤ ≤ x x 3 1 , or the interval [ ] −3, 1 . So the range of f is the interval [ ] −3, 1 . The range of −f 1 equals the domain of f. So the range of −f 1 is π π ⎡ − ⎣ ⎢ ⎤ ⎦ ⎥ 2 , 2 . Multiply by 2. Subtract 1 from each part. Interchange x and y. Solve for y. Definition of inverse sine function Now Work PROBLEM 65 9 Solve Equations Involving Inverse Trigonometric Functions Equations that contain inverse trigonometric functions are called inverse trigonometric equations. Solving an Inverse Trigonometric Equation Solve the equation: π = − x 3sin 1 EXAMPLE 11 Solution To solve an equation involving a single inverse trigonometric function, first isolate the inverse trigonometric function. π π = = − − x x 3sin sin 3 1 1 Divide both sides by 3.
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