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Stitz-Zeager_College_Algebra_e-book

# 87 2 2 and x 32 1 arcsin087 by graphing y sin2x

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Unformatted text preview: ction, we see that J (x) = tan(x) is undeﬁned at π x = π and x = 32 , in accordance with our ﬁndings in Section 10.3.1. As x → π − , sin(x) → 1− 2 2 sin( and cos(x) → 0+ , so that tan(x) = cos(x) → ∞ producing a vertical asymptote at x = π . Using a 2 x) − + π π similar analysis, we get that as x → π + , tan(x) → −∞, as x → 32 , tan(x) → ∞, and as x → 32 , 2 tan(x) → −∞. Plotting this information and performing the usual ‘copy and paste’ produces: y x 0 tan(x) (x, tan(x)) 0 (0, 0) π 4,1 π 4 π 2 3π 4 undeﬁned π 0 (π, 0) 5π 4 3π 2 7π 4 1 5π 4 ,1 undeﬁned 2π 0 1 −1 −1 1 3π 4 , −1 π 4 π 2 3π 4 π 5π 4 3π 2 7π 4 2π −1 7π 4 , −1 (2π, 0) The graph of y = tan(x) over [0, 2π ]. y x The graph of y = tan(x). x 10.5 Graphs of the Trigonometric Functions 687 From the graph, it appears as if the tangent function is periodic with period π . To prove that this is the case, we appeal to the sum formula for tangents. We have: tan(x + π ) = tan(x) + tan(π ) tan(x) + 0 = = tan(x), 1 − tan...
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