SECTION 6.4 Graphs of the Sine and Cosine Functions 427 OBJECTIVES 1 Graph the Sine Function = y x sin and Functions of the Form ω( ) = y A x sin (p. 427) 2 Graph the Cosine Function = y x cos and Functions of the Form ω( ) = y A x cos (p. 429) 3 Determine the Amplitude and Period of Sinusoidal Functions (p. 430) 4 Graph Sinusoidal Functions Using Key Points (p. 432) 5 Find an Equation for a Sinusoidal Graph (p. 436) 6.4 Graphs of the Sine and Cosine Functions • Graphing Techniques: Transformations (Section 2.5, pp. 112–120) Now Work the ‘Are You Prepared?’ problems on page 436. PREPARING FOR THIS SECTION Before getting started, review the following: We want to graph the trigonometric functions in the xy -plane. So we use the traditional symbols x for the independent variable (or argument) and y for the dependent variable for each function. Then the six trigonometric functions are written as ( ) ( ) ( ) ( ) ( ) ( ) = = = = = = = = = = = = y f x x y f x x y f x x y f x x y f x x y f x x sin cos tan csc sec cot Here the independent variable x represents an angle, measured in radians. However, in calculus, x will usually be treated as a real number. As noted earlier, these are equivalent ways of viewing x . 1 Graph the Sine Function = y x sin and Functions of the Form ω = ( ) y A x sin Because the sine function has period π2 , it is only necessary to graph = y x sin on the interval π [ ] 0,2 . The remainder of the graph will consist of repetitions of this portion of the graph. To begin, consider Table 6, which lists some points on the graph of = y x sin , for π ≤ ≤ x 0 2 . As the table shows, the graph of π = ≤ ≤ y x x sin , 0 2 , begins at the origin.As x increases from 0 to π 2 , the value of = y x sin increases from 0 to 1; as x increases from π 2 to π to π3 2 , the value of y decreases from 1 to 0 to −1; as x increases from π3 2 to π2 , the value of y increases from −1 to 0. Plotting the points listed in Table 6 and connecting them with a smooth curve yields the graph shown in Figure 47. Figure 47 π = ≤ ≤ y x x sin ,0 2 p–– 6 x y 1 p 2p 21 ( , )1 – 2 ( , ) 1 – 2 ( , 1) (0, 0) (p, 0) (2p, 0) p–– 2 5p––– 6 ( , 2 )1 – 2 7p––– 6 ) ( 2, 1 – 2 11p –––– 6 ( , 21) 3p––– 2 p –– 2 3p ––– 2 x sin = y x , ( ) x y 0 0 ( ) 0,0 π 6 1 2 π( ) 6 1 2 , π 2 1 π( ) 2 1, π5 6 1 2 π ( ) 5 6 1 2 , π 0 π( )0, π7 6 − 1 2 π ( ) − 7 6 1 2 , π3 2 −1 π ( ) − 3 2 1 , π 11 6 − 1 2 π ( ) − 11 6 1 2 , π2 0 π ( ) 2 0, Table 6
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