Class 11 Maths Chapter 3 Trigonometric Functions – Exercise 3.1 NCERT Solutions

Introduction

Exercise 3.1 introduces the basic trigonometric functions and their values at standard angles. You will learn how to define trigonometric ratios using a unit circle, evaluate trigonometric functions at specific angles, and understand their signs in different quadrants.

Key Concepts

1. Trigonometric Ratios:

$$\sin \theta =\frac{\text{Perpendicular}}{\text{Hypotenuse}},\cos \theta =\frac{\text{Base}}{\text{Hypotenuse}},\tan \theta =\frac{\text{Perpendicular}}{\text{Base}}$$

$$\csc \theta =\frac{1}{\sin \theta },\sec \theta =\frac{1}{\cos \theta },\cot \theta =\frac{1}{\tan \theta }$$

2. Signs of Trigonometric Functions (ASTC Rule):

   • I Quadrant: All positive

   • II Quadrant: Sine and cosec positive

   • III Quadrant: Tan and cot positive

   • IV Quadrant: Cos and sec positive

3. Standard Values:

$$\sin 0^{\circ }=0,\sin {30}^{\circ }=\frac{1}{2},\sin {45}^{\circ }=\frac{1}{\sqrt{2}},\sin {60}^{\circ }=\frac{\sqrt{3}}{2},\sin {90}^{\circ }=1$$

Similar values for $\cos \theta ,\tan \theta$.

NCERT Questions with Solutions (10)

Q1. Find $\sin {30}^{\circ },\cos {60}^{\circ },\tan {45}^{\circ }$.

$$\sin {30}^{\circ }=\frac{1}{2},\cos {60}^{\circ }=\frac{1}{2},\tan {45}^{\circ }=1$$

Q2. Find $\sin 0^{\circ },\cos {90}^{\circ },\tan 0^{\circ }$.

$$\sin 0^{\circ }=0,\cos {90}^{\circ }=0,\tan 0^{\circ }=0$$

Q3. Find $\sin {45}^{\circ },\cos {45}^{\circ }$.

$$\sin {45}^{\circ }=\cos {45}^{\circ }=\frac{1}{\sqrt{2}}$$

Q4. Find $\tan {30}^{\circ },\cot {60}^{\circ }$.

$$\tan {30}^{\circ }=\frac{1}{\sqrt{3}},\cot {60}^{\circ }=\frac{1}{\sqrt{3}}$$

Q5. Find $\sin {60}^{\circ },\cos {30}^{\circ }$.

$$\sin {60}^{\circ }=\frac{\sqrt{3}}{2},\cos {30}^{\circ }=\frac{\sqrt{3}}{2}$$

Q6. Find $\sec 0^{\circ },\csc {90}^{\circ }$.

$$\sec 0^{\circ }=1,\csc {90}^{\circ }=1$$

Q7. Find $\tan {60}^{\circ },\cot {30}^{\circ }$.

$$\tan {60}^{\circ }=\sqrt{3},\cot {30}^{\circ }=\sqrt{3}$$

Q8. Find $\sin {90}^{\circ },\cos 0^{\circ }$.

$$\sin {90}^{\circ }=1,\cos 0^{\circ }=1$$

Q9. Find $\tan {90}^{\circ }$. Undefined, since denominator = 0.

Q10. Find $\sec {90}^{\circ },\csc 0^{\circ }$.

$$\sec {90}^{\circ }=\text{undefined},\csc 0^{\circ }=\text{undefined}$$

FAQs (10)

1. What are trigonometric ratios? Ratios of sides of right triangle.

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

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. Which functions are undefined at 0°? $\csc 0^{\circ },\cot 0^{\circ }$.

10. Why is this exercise important? It builds foundation for advanced trigonometric identities.

Conclusion

Exercise 3.1 has 10 solved questions and 10 FAQs that strengthen your understanding of basic trigonometric functions and their values at standard angles. This builds the foundation for advanced trigonometric identities in Class 11 Maths.

visit:www.fuzymathacademy.com