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3.5 Inverse Trigonometric Functions

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Inverse sine (or arcsine) function f(x)=sin x is not one-to-one But the function f(x)=sin x, -π/2 ≤ x ≤ π/2 is one-to- one. The restricted sine function has an inverse function which is denoted by sin -1 or arcsin and is called inverse sine (or arcsine) function. Example: sin -1 (1/2) = π/6. Cancellation equations for sin and sin -1 :

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We can use implicit differentiation to find:

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But so is positive.

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Inverse cosine function f(x)=cos x is not one-to-one But the function f(x)=cos x, 0 ≤ x ≤ π is one-to-one. The restricted cosine function has an inverse function which is denoted by cos -1 or arccos and is called inverse cosine function. Example: cos -1 (1/2) = π/3. Cancellation equations for cos and cos -1 : Derivative of cos -1 :

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Inverse tangent function f(x)=tan x is not one-to-one But the function f(x)=tan x, -π/2 < x < π/2 is one-to- one. The restricted tangent function has an inverse function which is denoted by tan -1 or arctan and is called inverse tangent function. Example: tan -1 (1) = π/4. Limits involving tan -1 : Derivative of tan -1 :

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