Inverse trigonometric functions, also known as arcus functions, cyclometric functions, or anti-trigonometric functions, play a crucial role in various fields such as engineering, geometry, and navigation. They help us determine the angle corresponding to a given trigonometric value.

Notation for Inverse Trigonometric Functions:

The inverse trigonometric functions are typically denoted by prefixing 'arc' to the trigonometric function or by raising the function to the power of -1. For instance, the inverse of sin x can be written as:

arcsin(x) or

Now, let's dive into the process of deriving the inverse of a trigonometric function.

Example: Let's consider the function

Given the function

We can rewrite it as

Differentiating with respect to x gives us:

Substituting the value of y from equation (i), we get

It's clear from equation (ii) that cos y cannot be 0, as it would make the function undefined.

Which implies

From equation (i), we have

Using the property of trigonometric functions, we get

Substituting the value from equation (iii) into equation (ii), we get

Hence, the derivative of the inverse sine function is

Derivatives of Other Inverse Trigonometric Functions