NCERT Solutions Class 12 Maths Chapter 2 helps you master Inverse Trigonometric Functions through detailed solutions to all 43 questions across 3 exercises. You’ll learn how to find principal values of inverse trigonometric functions, simplify complex expressions using properties and identities, prove equations involving sin⁻¹, cos⁻¹, tan⁻¹, and solve problems that frequently appear in CBSE board exams. Each solution includes domain-range concepts, graphical interpretations, and shortcut techniques to tackle tricky questions confidently.
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All exercises with step-by-step solutions | Updated 2025-26 | Free Download
Download PDF (Free)NCERT Solutions Class 12 Maths Chapter 2 Inverse Trigonometric Functions – Complete Guide
NCERT Class 12 Chapter 2 on Inverse Trigonometric Functions introduces you to one of the most crucial topics in calculus and higher mathematics. You’ll explore the inverse functions of sine, cosine, tangent, cotangent, secant, and cosecant, understanding how these functions reverse the effect of their corresponding trigonometric functions. This chapter builds directly on your knowledge from Class 11 trigonometry and prepares you for integration and differentiation in later chapters.
📊 CBSE Class 12 Maths Chapter 2 – Exam Weightage & Marking Scheme
| CBSE Board Marks | 4 Marks |
| Unit Name | Relations and Functions |
| Difficulty Level | Medium |
| Importance | Medium |
| Exam Types | CBSE Board, State Boards |
| Typical Questions | 1-2 questions |
You will learn to determine the principal value branches of inverse trigonometric functions and understand why restricting domains is essential for these functions to exist. The chapter covers important properties like sin⁻¹(sin x) = x under specific conditions, and you’ll practice simplifying complex expressions involving multiple inverse functions. You’ll also master the addition and subtraction formulas for inverse trigonometric functions, which are frequently tested in CBSE board exams.
For CBSE Class 12 boards, this chapter carries a weightage of 4 marks and typically appears as 2-mark or 4-mark questions. You can expect questions asking you to find principal values, simplify expressions, prove identities, or solve equations involving inverse trigonometric functions. The medium difficulty level means questions require conceptual clarity and careful attention to domain restrictions.
Quick Facts – Class 12 Chapter 2
| 📖 Chapter Number | Chapter 2 |
| 📚 Chapter Name | Inverse Trigonometric Functions |
| ✏️ Total Exercises | 3 Exercises |
| ❓ Total Questions | 43 Questions |
| 📅 Updated For | CBSE Session 2025-26 |
Mastering inverse trigonometric functions is essential not just for board exams but also for competitive examinations like JEE and other engineering entrance tests. The concepts you learn here form the foundation for understanding definite integrals and differential equations. With consistent practice of NCERT solutions and previous years’ CBSE questions, you’ll develop the confidence to tackle any problem from this chapter efficiently.
NCERT Solutions Class 12 Maths Chapter 2 – All Exercises PDF Download
Download exercise-wise NCERT Solutions PDFs for offline study
| Exercise No. | Topics Covered | Download PDF |
|---|---|---|
| EXERCISE 2.1 | Complete step-by-step solutions for 14 questions | 📥 Download PDF |
| EXERCISE 2.2 | Complete step-by-step solutions for 15 questions | 📥 Download PDF |
| Miscellaneous Exercise on Chapter 2 | Complete step-by-step solutions for 14 questions | 📥 Download PDF |
Inverse Trigonometric Functions – Key Formulas & Concepts
Quick reference for CBSE exams
| Formula | Description | When to Use |
|---|---|---|
| Principal Value Range – sin⁻¹x \(-\frac{\pi}{2} \leq \sin^{-1}x \leq \frac{\pi}{2}\) | The range of the principal value of sin⁻¹x Note: Remember to convert your angle to radians if needed. sin⁻¹x gives an angle in the range [-π/2, π/2]. | Finding the principal value of sin⁻¹x, checking if the answer is within the valid range. |
| Principal Value Range – cos⁻¹x \(0 \leq \cos^{-1}x \leq \pi\) | The range of the principal value of cos⁻¹x Note: cos⁻¹x gives an angle in the range [0, π]. | Finding the principal value of cos⁻¹x, checking if the answer is within the valid range. |
| Principal Value Range – tan⁻¹x \(-\frac{\pi}{2} < \tan^{-1}x < \frac{\pi}{2}\) | The range of the principal value of tan⁻¹x Note: Note that the range is open, meaning -π/2 and π/2 are not included. tan⁻¹x gives an angle in the range (-π/2, π/2). | Finding the principal value of tan⁻¹x, checking if the answer is within the valid range. |
| sin⁻¹(sin x) \(\sin^{-1}(\sin x) = x\) | Simplifies sin⁻¹(sin x) to x, but only if x is within the principal value range. Note: This is only true if \(x \in [-\frac{\pi}{2}, \frac{\pi}{2}]\). If not, you need to adjust x to be within this range. | Simplifying expressions, solving equations involving inverse trigonometric functions. |
| cos⁻¹(cos x) \(\cos^{-1}(\cos x) = x\) | Simplifies cos⁻¹(cos x) to x, but only if x is within the principal value range. Note: This is only true if \(x \in [0, \pi]\). If not, you need to adjust x to be within this range. | Simplifying expressions, solving equations involving inverse trigonometric functions. |
| tan⁻¹(tan x) \(\tan^{-1}(\tan x) = x\) | Simplifies tan⁻¹(tan x) to x, but only if x is within the principal value range. Note: This is only true if \(x \in (-\frac{\pi}{2}, \frac{\pi}{2})\). If not, you need to adjust x to be within this range. | Simplifying expressions, solving equations involving inverse trigonometric functions. |
| sin⁻¹(-x) \(\sin^{-1}(-x) = -\sin^{-1}(x)\) | Handles the negative argument inside sin⁻¹ Note: Negative sign comes out directly | Simplifying expressions with negative arguments |
| tan⁻¹(-x) \(\tan^{-1}(-x) = -\tan^{-1}(x)\) | Handles the negative argument inside tan⁻¹ Note: Negative sign comes out directly | Simplifying expressions with negative arguments |
| cos⁻¹(-x) \(\cos^{-1}(-x) = \pi – \cos^{-1}(x)\) | Handles the negative argument inside cos⁻¹ Note: Important to remember the π subtraction | Simplifying expressions with negative arguments |
| sin⁻¹x + cos⁻¹x \(\sin^{-1}x + \cos^{-1}x = \frac{\pi}{2}\) | Relationship between sin⁻¹x and cos⁻¹x Note: This is a very useful identity to simplify expressions. | Simplifying expressions, solving equations |
| tan⁻¹x + cot⁻¹x \(\tan^{-1}x + \cot^{-1}x = \frac{\pi}{2}\) | Relationship between tan⁻¹x and cot⁻¹x Note: Similar to the sin⁻¹x + cos⁻¹x identity. | Simplifying expressions, solving equations |
| tan⁻¹x + tan⁻¹y \(\tan^{-1}x + \tan^{-1}y = \tan^{-1}\left(\frac{x+y}{1-xy}\right)\) | Sum of two tan⁻¹ functions Note: Valid when xy < 1. Pay attention to the condition for the formula to be valid. | Simplifying expressions involving sum of two inverse tangents |
| tan⁻¹x – tan⁻¹y \(\tan^{-1}x – \tan^{-1}y = \tan^{-1}\left(\frac{x-y}{1+xy}\right)\) | Difference of two tan⁻¹ functions Note: Valid when xy > -1. Pay attention to the condition for the formula to be valid. | Simplifying expressions involving difference of two inverse tangents |
| 2tan⁻¹x (in terms of sin⁻¹) \(2\tan^{-1}x = \sin^{-1}\left(\frac{2x}{1+x^2}\right)\) | Expressing 2tan⁻¹x as sin⁻¹ Note: Useful when you need to express 2tan⁻¹x in terms of sin⁻¹ for simplification or calculation. | Converting between inverse trigonometric functions, simplifying expressions. |
| 2tan⁻¹x (in terms of cos⁻¹) \(2\tan^{-1}x = \cos^{-1}\left(\frac{1-x^2}{1+x^2}\right)\) | Expressing 2tan⁻¹x as cos⁻¹ Note: Useful when you need to express 2tan⁻¹x in terms of cos⁻¹ for simplification or calculation. | Converting between inverse trigonometric functions, simplifying expressions. |
Frequently Asked Questions – NCERT Class 12 Maths Chapter 2
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