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Stitz-Zeager_College_Algebra_e-book

# 721 f tanx 06109 c cosx 09824 g tan x

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Unformatted text preview: Section 6 6 6 5.2 when we saw (−2)2 = 2 as opposed to −2. Additionally, even though the expression 1 − 2x2 is deﬁned for all real numbers, the equivalence cos (2 arcsin(x)) = 1 − 2x2 is valid for only −1 ≤ x ≤ 1. This is akin to the fact that while the expression x is deﬁned for all real numbers, the equivalence √2 ( x) = x is valid only for x ≥ 0. 3 In other words, the angle θ = t radians is a Quadrant I or II angle where sine is nonnegative. Alternatively, we could use the identity: 1 + tan2 (t) = sec2 (t). Since we are given x = cos(t), we know sec(t) = 1 1 = x . The reader is invited to work through this approach to see what, if any, diﬃculties arise. cos(t) 4 10.6 The Inverse Trigonometric Functions 705 The next pair of functions we wish to discuss are the inverses of tangent and cotangent. First, we restrict f (x) = tan(x) to its fundamental cycle on − π , π to obtain f −1 (x) = arctan(x). Among 22 other things, note that the vertical asymptotes x = − π and x = π of the graph of f (x) = tan(x) 2 2 become the horizontal asymptotes y = − π and y = π of the graph of f −1 (x) = arctan(x). 2 2 y y 1 π 2 −π −π 2 4 π 4 π 2 x π 4 −1 −1 x 1 −π 4 reﬂect across y = x f (x) = tan(x), − π < x < 2 −− − − − −→ −−−−−− π . 2 switch x and y coordinates −π 2 f −1 (x) = arctan(x). Next, we restrict g (x) = cot(x) to its fundamental cycle on (0, π ) to ob...
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