SECTION 7.1 The Inverse Sine, Cosine, and Tangent Functions 481 Figure 18 Now Work PROBLEM 55 For the cosine function and its inverse, the following properties hold. π ( ) ( ) ( ) = = ≤ ≤ − − f f x x x x cos cos where0 1 1 (5a) ( ) ( ) ( ) = = − ≤ ≤ − − f f x x x x cos cos where1 1 1 1 (5b) π π ( ) ( ) ( ) = = − < < − − f f x x x x tan tan where 2 2 1 1 ( ) ( ) ( ) = = −∞< <∞ − − f f x x x x tan tan where 1 1 Using Properties of Inverse Functions to Find the Exact Value of Certain Composite Functions Find the exact value, if any, of each composite function. (a) π ( ) − cos cos 12 1 (b) [ ] ( ) − − cos cos 0.4 1 (c) π ( ) ⎡ − ⎣ ⎢ ⎤ ⎦ ⎥ − cos cos 2 3 1 (d) π ( ) − cos cos 1 Solution EXAMPLE 9 (a) π π ( ) = − cos cos 12 12 1 12 π is in the interval 0, π [ ] ; use property (5a). (b) [ ] ( ) − = − − cos cos 0.4 0.4 1 0.4 − is in the interval 1, 1 ; [ ] − use property (5b). (c) The angle π − 2 3 is not in the interval π [ ] 0, , so property (5a) cannot be used. However, because the cosine function is even, π π ( ) − = cos 2 3 cos 2 3 . Because π2 3 is in the interval π [ ] 0, , property (5a) can be used, and π π π ( ) ( ) ⎡ − ⎣ ⎢ ⎤ ⎦ ⎥ = = − − cos cos 2 3 cos cos 2 3 2 3 1 1 (d) Because π is not in the interval [ ] −1, 1 , the domain of the inverse cosine function, π − cos 1 is not defined. This means the composite function cos π ( ) − cos 1 is also not defined. Now Work PROBLEMS 39 AND 59 For the tangent function and its inverse, the following properties hold. Now Work PROBLEM 47
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