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3.5 Inverse Trigonometric Functions. Inverse sine (or arcsine) function. f(x)=sin x is not one-to-one
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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/2is 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:
But so is positive. We can use implicit differentiation to find:
Inverse cosine function • f(x)=cos x is not one-to-one • But the function f(x)=cos x , 0 ≤ x ≤ πis 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 :
Inverse tangent function • f(x)=tan x is not one-to-one • But the function f(x)=tan x , -π/2 < x < π/2is 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:
