Class 12 Maths Chapter 2 Inverse Trigonometric Functions – Exercise 2.1 NCERT Solutions

Introduction

Exercise 2.1 introduces inverse trigonometric functions. Students learn how to define principal values, evaluate expressions, and simplify problems involving inverse trigonometric identities. This exercise is foundational for calculus, integration, and advanced trigonometry.

Formulas Used

1. Principal Values:

$${\sin }^{-1}x\in [-\frac{\pi }{2},\frac{\pi }{2}],{\cos }^{-1}x\in [0,\pi ],{\tan }^{-1}x\in (-\frac{\pi }{2},\frac{\pi }{2})$$

2. Basic Identities:

$${\sin }^{-1}x+{\cos }^{-1}x=\frac{\pi }{2},{\tan }^{-1}x+{\cot }^{-1}x=\frac{\pi }{2}$$

3. Conversion:

$${\tan }^{-1}\left(\frac{1}{x}\right)={\cot }^{-1}x,x>0$$

Students Frequently Make Mistakes

• Forgetting principal value ranges.

• Confusing ${\sin }^{-1}x$ with $\frac{1}{\sin x}$.

• Misapplying identities without checking domain restrictions.

• Errors in simplification of composite inverse functions.

• Ignoring quadrant rules in evaluation.

NCERT Questions with Step‑by‑Step Solutions (10)

Q1. Evaluate ${\sin }^{-1}\left(\frac{\sqrt{3}}{2}\right)$.

$${\sin }^{-1}\left(\frac{\sqrt{3}}{2}\right)=\frac{\pi }{3}$$

Q2. Evaluate ${\cos }^{-1}\left(\frac{1}{2}\right)$.

$${\cos }^{-1}\left(\frac{1}{2}\right)=\frac{\pi }{3}$$

Q3. Evaluate ${\tan }^{-1}(1)$.

$${\tan }^{-1}(1)=\frac{\pi }{4}$$

Q4. Evaluate ${\cot }^{-1}(1)$.

$${\cot }^{-1}(1)=\frac{\pi }{4}$$

Q5. Evaluate ${\sec }^{-1}(2)$.

$${\sec }^{-1}(2)=\frac{\pi }{3}$$

Q6. Evaluate ${\csc }^{-1}\left(\frac{2}{\sqrt{3}}\right)$.

$${\csc }^{-1}\left(\frac{2}{\sqrt{3}}\right)=\frac{\pi }{3}$$

Q7. Simplify ${\sin }^{-1}x+{\cos }^{-1}x$.

$${\sin }^{-1}x+{\cos }^{-1}x=\frac{\pi }{2}$$

Q8. Simplify ${\tan }^{-1}x+{\cot }^{-1}x$.

$${\tan }^{-1}x+{\cot }^{-1}x=\frac{\pi }{2}$$

Q9. Evaluate ${\tan }^{-1}\left(\frac{1}{\sqrt{3}}\right)$.

$${\tan }^{-1}\left(\frac{1}{\sqrt{3}}\right)=\frac{\pi }{6}$$

Q10. Evaluate ${\sin }^{-1}(0)$.

$${\sin }^{-1}(0)=0$$

FAQs (10)

FAQ1. What is inverse trigonometric function? Inverse of trigonometric function with restricted domain.

FAQ2. What is principal value branch? Specific range chosen for inverse function.

FAQ3. What is range of ${\sin }^{-1}x$? $[-\frac{\pi }{2},\frac{\pi }{2}]$.

FAQ4. What is range of ${\cos }^{-1}x$? $[0,\pi ]$.

FAQ5. What is range of ${\tan }^{-1}x$? $(-\frac{\pi }{2},\frac{\pi }{2})$.

FAQ6. What is identity of ${\sin }^{-1}x+{\cos }^{-1}x$? $\frac{\pi }{2}$.

FAQ7. What is identity of ${\tan }^{-1}x+{\cot }^{-1}x$? $\frac{\pi }{2}$.

FAQ8. Is ${\sin }^{-1}x=\frac{1}{\sin x}$? No, it is inverse function, not reciprocal.

FAQ9. Why restrict domain of trigonometric functions? To make them one‑one and invertible.

FAQ10. Why is Exercise 2.1 important? It builds foundation for calculus and integration involving inverse trigonometric functions.

Conclusion

Exercise 2.1 has 10 solved questions and 10 FAQs that strengthen your understanding of inverse trigonometric functions, principal values, and identities. This builds the foundation for calculus in Class 12 Maths.

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