NCERT Class 12-Mathematics: Chapter –2 Inverse Trigonometric Function Part 1

Get top class preparation for NSO-Level-2 right from your home: fully solved questions with step-by-step explanation- practice your way to success.

2.1 Overview

2.1.1 Inverse function

Inverse of a function ‘ ‘ exists, if the function is one-one and onto, i.e., bijective. Since trigonometric functions are many-one over their domains, we restrict their domains and co-domains in order to make them one-one and onto and then find their inverse. The domains and ranges (principal value branches) of inverse trigonometric functions are given below:

Notes:

(i) The symbol sin–1x should not be confused is an angle, the value of whose sine is x, similarly for other trigonometric functions.

(ii) The smallest numerical value, either positive or negative, of is called the principal value of the function.

(iii) Whenever no branch of an inverse trigonometric function is mentioned, we mean the principal value branch. The value of the inverse trigonometric function which lies in the range of principal branch is its principal value.

2.1.2 Graph of an inverse trigonometric function

The graph of an inverse trigonometric function can be obtained from the graph of original function by interchanging and , i.e., if is a point on the graph of trigonometric function, then becomes the corresponding point on the graph of its inverse trigonometric function.

It can be shown that the graph of an inverse function can be obtained from the corresponding graph of original function as a mirror image (i.e., reflection) along the line .

2.1.3 Properties of inverse trigonometric functions

1.

2.

3.

4.

5.

6.

7.

2.2 Solved Examples

Short Answer (S.A)

Question 1:Find the principal value of , for

Answer:

If , then .

Since we are considering principal branch, . Also, since being in the first quadrant, hence.