480 CHAPTER 7 Analytic Trigonometry (b) The composite function π ( ) − sin sin 5 8 1 follows the form of property (4a). But because π5 8 is not in the interval π π π π ( ) ⎡ − ⎣ ⎢ ⎤ ⎦ ⎥ ≠ − 2 , 2 , sin sin 5 8 5 8 . 1 To find π ( ) − sin sin 5 8 , 1 first find an angle θ in the interval π π ⎡ − ⎣ ⎢ ⎤ ⎦ ⎥ 2 , 2 for which θ π = sin sin 5 8 . Figure 15 illustrates that π π = = y sin 5 8 sin 3 8 . Since π3 8 is in the interval π π ⎡ − ⎣ ⎢ ⎤ ⎦ ⎥ 2 , 2 , we can use (4a). π π π ( ) ( ) = = − − sin sin 5 8 sin sin 3 8 3 8 1 1 ↑ Use property (4a). These results can be verified with technology. Using Desmos, Figure 16(a) shows that π π ( ) ≠ − sin sin 5 8 5 8 . 1 Figure 16(b) confirms that π π π ( ) ( ) = = − − sin sin 5 8 sin sin 3 8 3 8 1 1 Figure 16 (a) (b) Figure 15 sin 5 8 sin 3 8 π π = y x x2 1 y2 5 1 (2x, y) (x, y) 1 21 21 1 3p ––– 8 3p ––– 8 5p ––– 8 Now Work PROBLEM 43 Finding the Exact Value of Certain Composite Functions Find the exact value, if any, of each composite function. (a) ( ) − sin sin 0.8 1 (b) ( ) − sin sin 1.3 1 EXAMPLE 8 Solution (a) The composite function ( ) − sin sin 0.8 1 follows the form of property (4b), and 0.8 is in the interval [ ] −1, 1 . Using (4b) reveals that ( ) = − sin sin 0.8 0.8 1 Figure 17 verifies the result using a TI-84 Plus CE graphing calculator. (b) The composite function ( ) − sin sin 1.3 1 follows the form of property (4b). But since 1.3 is not in the domain of the inverse sine function, [ ] − − 1, 1 , sin 1.3 1 is not defined. Therefore, ( ) − sin sin 1.3 1 is also not defined. See Figure 18 on the next page. Figure 17
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