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Class XII Mathematics

Chapter 2: Inverse Trigonometric Functions

Standard NCERT & CBSE aligned study curriculum. Master concepts, track accuracy, revise weak areas, and challenge yourself with 9 customized practice modes.

Chapter Overview

Welcome to Class XII Mathematics: Inverse Trigonometric Functions. This chapter forms a core structural component of the math syllabus, designed to build analytical rigor and key formula models.

Use the detailed subtopic guide below to review standard definitions, key mathematical rules, and study guidelines.

Prerequisite Concepts

Trigonometric FunctionsRelations and Functions

Detailed Subtopics Study Guide

Review detailed conceptual explanations, mathematical equations, and guidelines for each subtopic in this chapter:

1Definition, domain, and range

Concept Explanation

Trigonometric functions are periodic and not 1-to-1. To define their inverses, we restrict their domains to specific intervals, creating principal value branches where the functions are bijective.

Mathematical Representation
\sin^{-1}: [-1, 1] \to [-\pi/2, \pi/2], \quad \cos^{-1}: [-1, 1] \to [0, \pi], \quad \tan^{-1}: \mathbb{R} \to (-\pi/2, \pi/2)
Study Guideline: Always ensure the output value lies within the designated Principal Value Branch range for that specific inverse function.

2Principal value branches

Concept Explanation

The principal value branch is the standard range of outputs defined for inverse trigonometric functions. Any evaluation of an inverse trig function must yield a value in this branch.

Mathematical Representation
\theta = \sin^{-1}(x) \implies -\frac{\pi}{2} \le \theta \le \frac{\pi}{2}
Study Guideline: For negative inputs, remember: sin⁻¹(-x) = -sin⁻¹(x), whereas cos⁻¹(-x) = π - cos⁻¹(x).

3Graphs of inverse trigonometric functions

Concept Explanation

The graph of an inverse trigonometric function is the reflection of the restricted trigonometric graph across the line y = x. The roles of the horizontal and vertical axes are swapped.

Mathematical Representation
x = \sin y \iff y = \sin^{-1} x \quad \text{plotted with } x \in [-1, 1]
Study Guideline: The horizontal axis represents the numerical values, and the vertical axis represents the angles (in radians).

4Properties of inverse trig functions

Concept Explanation

Properties of inverse trigonometric functions simplify complex expressions, including reciprocal relationships and sum/difference identities.

Mathematical Representation
\sin^{-1} x + \cos^{-1} x = \frac{\pi}{2}, \quad \tan^{-1} x + \tan^{-1} y = \tan^{-1}\left(\frac{x+y}{1-xy}\right)
Study Guideline: When applying the tan⁻¹(x) + tan⁻¹(y) formula, ensure that the product xy is strictly less than 1.