645 6.5 Graphs of the Tangent and Cotangent Functions x y =tan x - p 3 - 23 ≈ -1.7 - p 4 -1 - p 6 - 23 3 ≈ -0.6 0 0 p 6 23 3 ≈0.6 p 4 1 p 3 23 ≈1.7 p 0 –2 2 y = tan x y –P 2 –p 6 –p 3 p 3 p 6 P 2 x –p –p 4 p 4 Figure 41 The tangent function has period p. Because tan x = sin x cos x , tangent values are 0 when sine values are 0, and are undefined when cosine values are 0. As x-values increase from - p 2 to p 2 , tangent values range from -∞ to ∞ and increase throughout the interval. Those same values are repeated as x increases from p 2 to 3p 2 , from 3p 2 to 5p 2 , and so on. The graph of y = tan x from - 3p 2 to 3p 2 is shown in Figure 42. –2 0 2 y = tan x Period: p p –p y 3p 2 – –p 2 p 4 p 2 3p 2 –p 4 x The graph continues in this pattern. Figure 42 Tangent Function ƒ 1x2 =tan x Domain: Ex x ≠12n + 12 p 2 , where n is any integerF Range: 1-∞, ∞2 x y - p 2 undefined - p 4 -1 0 0 p 4 1 p 2 undefined Figure 43 –2 y 0 1 2 f(x) = tan x, – < x < x 2 p 2 p –p 2 –p 4 p 4 p 2 −4 4 11p 4 f(x) = tan x 11p 4 − • The graph is discontinuous at values of x of the form x = 12n + 12 p 2 and has vertical asymptotes at these values. • Its x-intercepts have x-values of the form np. • 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, tan1-x2 = -tan x. Graph of the Cotangent Function A similar analysis for selected points between 0 and p for the graph of the cotangent function yields the graph in Figure 44 on the next page. Here the vertical asymptotes are at x-values that are integer multiples of p. Because cot1-x2 = -cot x, Odd function the graph of the cotangent function is also symmetric with respect to the origin.
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