Class 12 Mathematics Chapter 2 Inverse Trigonometric Functions – Revision Notes

Inverse Trigonometric Functions restricts sin, cos, and tan to principal value branches so that each has a well-defined inverse, and this chapter lays out the key identities that follow from that restriction.

Last Updated: September 23, 2026

Principal Value Branches

• sin⁻¹x: [−1,1]→[−π/2,π/2]. cos⁻¹x: [−1,1]→[0,π]. tan⁻¹x: R→(−π/2,π/2).

Key Identities

• sin⁻¹x+cos⁻¹x=π/2; tan⁻¹x+cot⁻¹x=π/2.
• sin⁻¹(−x)=−sin⁻¹x; cos⁻¹(−x)=π−cos⁻¹x.

Sum Formula

• tan⁻¹x+tan⁻¹y=tan⁻¹((x+y)/(1−xy)), valid for xy<1.

One-Line Summary

Inverse trigonometric functions are defined by restricting trig functions to specific principal value domains where they become one-to-one and invertible, governed by a set of identities relating them to one another.

Frequently Asked Questions

Why do trigonometric functions need a restricted domain to have an inverse?
Trigonometric functions like sine and cosine are periodic and not one-one over their entire domain, so their inverses only exist when the domain is restricted to an interval where the function is one-one and onto, called the principal value branch.

What is the principal value branch of the inverse sine function?
The principal value branch of inverse sine is the interval from minus pi by 2 to pi by 2, which is the range of values the inverse sine function returns for any input between minus 1 and 1.

Written by Satish

NCERTBooks.org is an independent educational resource run by a small team focused on making official NCERT textbooks easy to find, read, and download for students, parents, and teachers across India. We are not affiliated with NCERT or the Ministry of Education -- we organise publicly available NCERT content by class and subject, verify links against official sources, and build tools (like our in-browser reader) that make studying more convenient. Every guide we publish is written and reviewed by our team based on the actual NCERT curriculum and syllabus.