The inverse Sine, Cosine, and Tangent Functions PDF

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The inverse Sine, Cosine, and Tangent Functions...

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The inverse Sine, Cosine, and Tangent Functions : 10/01/2012 The inverse sine function

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The equation for the inverse of y = f (x) = sinx is obtained by interchanging x and y. ⇒ y = sin−1 x means x = siny , where −1 ≤ x ≤ 1 and −π ≤ y ≤ π2 2 y is inverse sine of x Example: Find sin−1 (− 12 ) , sin−1 ( 31 ) Use properties of Inverse functions to find exact values of certain composite functions : We know f −1 (f (x)) = x ∴ sin−1 (sin(x)) = x, where −π ≤ x ≤ π2 2 and sin(sin−1 (x)) = x, where −1 ≤ x ≤ 1 Examples : Find sin−1 (sin π8 ) and sin(sin−1 (1.8)) The inverse Cosine function : y = cos−1 x means x = cosy Where −1 ≤ x ≤ 1 and√0 ≤ y ≤ π Examples: cos−1 ( −2 2 ) Properties of inverse function : cos−1 (cosx) = x, where 0 ≤ x ≤ π cos(cos−1 x), where −1 ≤ x ≤ 1 )) Examples : cos(cos−1 (−0.4)) and cos−1 (cos(− 2π 3

The inverse Tangent function

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y = tan−1 x means x = tany where −∞ < x < ∞ and − π2 ≤ y ≤

π 2

Examples: Find tan−1 1 Properties of inverse function: ≤ x ≤ 2π tan−1 (tanx) = x, where −π 2 −1 tan(tan x) = x, where −∞ < x < ∞ 1

Find inverse function of trigonometric function

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Examples : Find f −1 of f (x) = 2sinx − 1, − π2 ≤ x ≤ and domain and range of f −1

π. 2

Find range of f

Solve equations involving inverse trigonometric functions Example : 3sin−1 = π

2

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