3.5 Inverse Trigonometric Functions

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:

We can use implicit differentiation to find:

We can use implicit differentiation to find: But so is positive.

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 :

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: