620 CHAPTER 6 The Circular Functions and Their Graphs 6.3 Graphs of the Sine and Cosine Functions ■ Periodic Functions ■ Graph of the Sine Function ■ Graph of the Cosine Function ■ Techniques for Graphing, Amplitude, and Period ■ Connecting Graphs with Equations ■ ATrigonometric Model Periodic Functions Phenomena that repeat with a predictable pattern, such as tides, phases of the moon, and hours of daylight, can be modeled by sine and cosine functions. These functions are periodic — that is, their values repeat in regular intervals, or periods. The periodic graph in Figure 20 represents a normal heartbeat. Figure 20 Periodic functions are defined as follows. Periodic Function A periodic function is a function ƒ such that ƒ1x2 =ƒ1x +np2, for every real number x in the domain of ƒ, every integer n, and some positive real number p. The least possible positive value of p is the period of the function. LOOKING AHEAD TO CALCULUS Periodic functions are used throughout calculus, so it is important to know their characteristics. One use of these functions is to describe the location of a point in the plane using polar coordinates, an alternative to rectangular coordinates. The circumference of the unit circle is 2p, so the least value of p for which the sine and cosine functions repeat is 2p. Therefore, the sine and cosine functions are periodic functions with period 2P. For every positive integer n, sin x =sin1x +n # 2P2 and cos x =cos1x +n # 2P2. Graph of the Sine Function We have seen that for a real number s, the point on the unit circle corresponding to s has coordinates 1cos s, sin s2. See Figure 21. Trace along the circle to verify the results shown in the table. x y (–1, 0) (1, 0) ( x, y) = (0, –1) (0, 1) 0 0 p (cos s, sin s) The unit circle x2 + y2 = 1 s 3p 2 p 2 Figure 21 As s Increases from sin s cos s 0 to p 2 Increases from 0 to 1 Decreases from 1 to 0 p 2 to p Decreases from 1 to 0 Decreases from 0 to -1 p to 3p 2 Decreases from 0 to -1 Increases from -1 to 0 3p 2 to 2p Increases from -1 to 0 Increases from 0 to 1 To avoid confusion when graphing the sine function, we use x rather than s. This corresponds to the letters in the xy-coordinate system. Selecting key values of x and finding the corresponding values of sin x leads to the table in Figure 22 on the next page.
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