646 CHAPTER 6 The Circular Functions and Their Graphs x y =cot x p 6 23 ≈1.7 p 4 1 p 3 23 3 ≈0.6 p 2 0 2p 3 - 23 3 ≈ -0.6 3p 4 -1 5p 6 - 23 ≈ -1.7 p 2 5p 6 p 6 3 p y = cot x p 4 2p 3 –2 –1 0 2 1 P x y 3p 4 Figure 44 The cotangent function also has period p. Cotangent values are 0 when cosine values are 0, and are undefined when sine values are 0. As x-values increase from 0 to p, cotangent values range from ∞ to -∞ and decrease throughout the interval. Those same values are repeated as x increases from p to 2p, from 2p to 3p, and so on. The graph of y = cot x from -p to p is shown in Figure 45. 0 1 x y y = cot x Period: p –p p –p 2 p 2 The graph continues in this pattern. Figure 45 Cotangent Function ƒ1x2 =cot x Domain: 5x x ≠np, where n is any integer6 Range: 1-∞, ∞2 x y 0 undefined p 4 1 p 2 0 3p 4 -1 p undefined Figure 46 f(x) = cot x, 0 < x < P 0 1 –1 y x p p 4 p 2 −4 4 11p 4 f(x) = cot x 11p 4 − • The graph is discontinuous at values of x of the form x = np and has vertical asymptotes at these values. • Its x-intercepts have x-values of the form 12n + 12 p 2 . • Its period is p. • 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, cot1-x2 = -cot x. The tangent function can be graphed directly with a graphing calculator, using the tangent key. To graph the cotangent function, however, we must use one of the identities cot x = 1 tan x or cot x = cos x sin x because graphing calculators generally do not have cotangent keys. 7
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