655 6.6 Graphs of the Secant and Cosecant Functions Graph of the Cosecant Function A similar analysis for selected points between -p and p for the graph of the cosecant function yields the graph in Figure 57. The vertical asymptotes are at x-values that are integer multiples of p. This graph is symmetric with respect to the origin because csc1-x2 = -csc x. Odd function –1 x 0 1 y = sin x y = csc x Period: 2p y –2p 2p –p p Figure 58 y x y = csc x –p 2 –P p 2 P –1 0 1 2 Figure 57 x y =csc x x y =csc x p 6 2 - p 62 p 4 22 ≈1.4 - p 4 -22 ≈ -1.4 p 3 223 3 ≈1.2 - p 3 - 223 3 ≈ -1.2 p 2 1 - p 21 2p 3 223 3 ≈1.2 - 2p 3 - 223 3 ≈ -1.2 3p 4 2 ≈1.4 - 3p 4 - 2 ≈ -1.4 5p 6 2 - 5p 62 Because cosecant values are reciprocals of corresponding sine values, the period of the cosecant function is 2p, the same as for y = sin x. When sin x = 1, the value of csc x is also 1. Likewise, when sin x = -1, csc x = -1. For all x, -1 … sin x … 1, and thus csc x Ú 1 for all x in its domain. Figure 58 shows how the graphs of y = sin x and y = csc x are related. –1 0 1 y f(x) = csc x x –2p 2p –p p −4 4 11p 4 f(x) = csc x 11p 4 − Figure 59 • The graph is discontinuous at values of x of the form x = np and has vertical asymptotes at these values. • There are no x-intercepts. • Its period is 2p. • There are no minimum or maximum values, so its graph has no amplitude. • The graph is symmetric with respect to the origin, so the function is an odd function. For all x in the domain, csc1-x2 = -csc x. Cosecant Function ƒ1x2 =csc x Domain: 5x x ≠np, where n is any integer6 Range: 1-∞, -14 ´31, ∞2 x y 0 undefined p 6 2 p 3 2 3 3 p 2 1 2p 3 223 3 p undefined 3p 21 2p undefined
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