1. 3.5 Inverse Trigonometric Functions

2. 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:

3. We can use implicit differentiation to find:

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

5. 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 :

6. 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: