Class 11 Maths Chapter 3 Trigonometric Functions – Exercise 3.4 NCERT Solutions

Introduction

Exercise 3.4 focuses on trigonometric functions of general angles. You will learn how to evaluate trigonometric functions for angles beyond the first quadrant, use the ASTC rule, and apply reduction formulas to simplify expressions. This exercise strengthens your ability to handle trigonometric values across all quadrants.

Key Concepts

1. ASTC Rule (Signs of Functions):

   • I Quadrant: All positive

   • II Quadrant: Sine and cosec positive

   • III Quadrant: Tan and cot positive

   • IV Quadrant: Cos and sec positive

2. Reduction Formulas:

$$\sin ({180}^{\circ }-\theta )=\sin \theta ,\cos ({180}^{\circ }-\theta )=-\cos \theta$$

$$\tan ({180}^{\circ }+\theta )=\tan \theta ,\cot ({180}^{\circ }+\theta )=\cot \theta$$

$$\sin ({360}^{\circ }-\theta )=-\sin \theta ,\cos ({360}^{\circ }-\theta )=\cos \theta$$

3. General Angle Values: Trigonometric functions repeat every ${360}^{\circ }$ or $2\pi$ radians.

NCERT Questions with Solutions (10)

Q1. Find $\sin {150}^{\circ }$.

$$\sin {150}^{\circ }=\sin ({180}^{\circ }-{30}^{\circ })=\sin {30}^{\circ }=\frac{1}{2}$$

Q2. Find $\cos {150}^{\circ }$.

$$\cos {150}^{\circ }=\cos ({180}^{\circ }-{30}^{\circ })=-\cos {30}^{\circ }=-\frac{\sqrt{3}}{2}$$

Q3. Find $\tan {210}^{\circ }$.

$$\tan {210}^{\circ }=\tan ({180}^{\circ }+{30}^{\circ })=\tan {30}^{\circ }=\frac{1}{\sqrt{3}}$$

Q4. Find $\cot {210}^{\circ }$.

$$\cot {210}^{\circ }=\cot ({180}^{\circ }+{30}^{\circ })=\cot {30}^{\circ }=\sqrt{3}$$

Q5. Find $\sin {330}^{\circ }$.

$$\sin {330}^{\circ }=\sin ({360}^{\circ }-{30}^{\circ })=-\sin {30}^{\circ }=-\frac{1}{2}$$

Q6. Find $\cos {330}^{\circ }$.

$$\cos {330}^{\circ }=\cos ({360}^{\circ }-{30}^{\circ })=\cos {30}^{\circ }=\frac{\sqrt{3}}{2}$$

Q7. Find $\tan {135}^{\circ }$.

$$\tan {135}^{\circ }=\tan ({180}^{\circ }-{45}^{\circ })=-\tan {45}^{\circ }=-1$$

Q8. Find $\cot {135}^{\circ }$.

$$\cot {135}^{\circ }=\cot ({180}^{\circ }-{45}^{\circ })=-\cot {45}^{\circ }=-1$$

Q9. Find $\sec {240}^{\circ }$.

$$\sec {240}^{\circ }=\sec ({180}^{\circ }+{60}^{\circ })=-\sec {60}^{\circ }=-2$$

Q10. Find $\csc {300}^{\circ }$.

$$\csc {300}^{\circ }=\csc ({360}^{\circ }-{60}^{\circ })=-\csc {60}^{\circ }=-\frac{2}{\sqrt{3}}$$

FAQs (10)

1. What is ASTC rule? Rule for signs of trigonometric functions in quadrants.

2. What is reduction formula? Formula to reduce angle to first quadrant.

3. What is domain of sine function? All real numbers.

4. What is range of sine function? $[-1,1]$.

5. What is domain of cosine function? All real numbers.

6. What is range of cosine function? $[-1,1]$.

7. What is domain of tangent function? All real numbers except odd multiples of $\frac{\pi }{2}$.

8. What is range of tangent function? $\mathbb{R}$.

9. Why use reduction formulas? To simplify trigonometric values of general angles.

10. Why is this exercise important? It builds skills in evaluating trigonometric functions beyond standard angles.

Conclusion

Exercise 3.4 has 10 solved questions and 10 FAQs that strengthen your understanding of trigonometric functions of general angles. This completes the Trigonometric Functions chapter in Class 11 Maths.

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