621 6.3 Graphs of the Sine and Cosine Functions To obtain the traditional graph in Figure 22, we plot the points from the table, use symmetry, and join them with a smooth curve. Because y = sin x is periodic with period 2p and has domain 1-∞, ∞2, the graph continues in the same pattern in both directions. This graph is a sine wave, or sinusoid. NOTE A function ƒ is an odd function if for all x in the domain of ƒ, ƒ1 −x2 = −ƒ1x2. The graph of an odd function is symmetric with respect to the origin. This means that if 1x, y2 belongs to the function, then 1-x, -y2 also belongs to the function. For example, Ap 2 , 1B and A - p 2 , -1B are points on the graph of y = sin x, illustrating the property sin1-x2 = -sin x. The sine function is related to the unit circle. Its domain consists of real numbers corresponding to angle measures (or arc lengths) on the unit circle. Its range corresponds to y-coordinates (or sine values) on the unit circle. Consider the unit circle in Figure 21 and assume that the line from the origin to some point on the circle is part of the pedal of a bicycle, with a foot placed on the circle itself. As the pedal is rotated from 0 radians on the horizontal axis through various angles, the angle (or arc length) giving the pedal’s location and its corresponding height from the horizontal axis given by sin x are used to create points on the sine graph. See Figure 23 on the next page. Sine Function ƒ 1x2 =sin x Domain: 1-∞, ∞2 Range: 3-1, 14 x y 0 0 p 6 1 2 p 4 22 2 p 3 23 2 p 2 1 p 0 3p 21 2p 0 • The graph is continuous over its entire domain, 1-∞, ∞2. • Its x-intercepts have x-values of the form np, where n is an integer. • Its period is 2p. • The graph is symmetric with respect to the origin, so the function is an odd function. For all x in the domain, sin1-x2 = -sin x. x y 2 1 0 –1 –2 f(x) = sin x, –2P " x " 2P –2p 2p –p p – – 3p 2 3p 2 p 2 p 2 f(x) = sin x −4 4 11p 4 11p 4 − Figure 22
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