432 CHAPTER 6 Trigonometric Functions One period of the graph of ω( ) = y x sin or ω( ) = y x cos is called a cycle . Figure 59 illustrates the general situation. The blue portion of the graph is one cycle. Recall A function f is even if ( ) ( ) − = f x f x ; a function f is odd if ( ) ( ) − =− f x f x . Since the sine function is odd, ( ) − =− x x sin sin ; since the cosine function is even, ( ) − = x x cos cos . j Finding the Amplitude and Period of a Sinusoidal Function Determine the amplitude and period of ( ) = y x 3sin 4 . EXAMPLE 4 Solution Comparing ( ) = y x 3sin4 to ω( ) = y A x sin , note that = A 3 and ω = 4. From equation (1), π ω π π = = = = = = A T Amplitude 3 Period 2 2 4 2 Figure 59 (a) ω ω π ω ( ) = > > = y A x A sin , 0, 0;period 2 x y A 2A p –– v 2p ––– v (b) ω ω π ω ( ) = > > = y A x A cos , 0, 0;period 2 x y A 2A p –– v 2p ––– v If ω > 0 and ω ( ) = − y x sin or ω ( ) = − y x cos , we use the even–odd properties of the sine and cosine functions, which are ω ω ω ω ( ) ( ) ( ) ( ) − = − − = x x x x sin sin and cos cos This gives us an equivalent form in which the coefficient of x is positive. For example, π π ( ) ( ) ( ) ( ) − = − − = x x x x sin 2 sin 2 and cos cos Because of this, we can assume that ω > 0. THEOREM Amplitude and Period If ω > 0, the amplitude and period of ω( ) = y A x sin and ω( ) = y A x cos are given by π ω = = = A T Amplitude Period 2 (1) Now Work PROBLEM 19 4 Graph Sinusoidal Functions Using Key Points So far, we have graphed functions of the form ω( ) = y A x sin or ω( ) = y A x cos using transformations. We now introduce another method that can be used to graph these functions. Figure 60 shows one cycle of the graphs of = y x sin and = y x cos on the interval π [ ] 0,2 . Each graph consists of four parts corresponding to the four subintervals: π π π π π π π ⎡ ⎣ ⎢ ⎤ ⎦ ⎥ ⎡ ⎣ ⎢ ⎤ ⎦ ⎥ ⎡ ⎣ ⎢ ⎤ ⎦ ⎥ ⎡ ⎣ ⎢ ⎤ ⎦ ⎥ 0, 2 2 , , 3 2 3 2 , 2 Each subinterval is of length π 2 (the period π2 divided by 4), and the endpoints of these intervals π π π π = = = = = x x x x x 0, 2 , , 3 2 , 2 give rise to five key points on each graph, as shown in Figure 60. Figure 60 x y (a) y 5 sin x 1 ( , 1) (0, 0) (p, 0) (2p, 0) p –– 2 ( , 21) 21 3p ––– 2 x y 1 (0, 1) (b) y 5 cos x (p, 21) (2p, 1) ( , 0) p –– 2 21 ( , 0) 3p ––– 2
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