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# NCERT Solutions for Class 11 Maths Chapter 3 Trigonometric Functions Exercise 3.3 - PDF Download

NCERT solutions for class 11 maths chapter 3 Trigonometric Functions exercise 3.3 is largely formula-based. In previous classes students have just learned to express a trigonometric function. New trigonometric formulas are derived using these fundamental concepts. You must not only understand the roots of these formulas but also use them to simplify difficult calculations. You also learn how to use the function-based formula to prove the trigonometric identity.

Ex 3.3 class 11 maths chapter 3 solutions comprises 25 questions on the trigonometric function of the sum and difference of two angles. Class 11 maths chapter 3 exercise 3.3 NCERT solutions provided by subject experts of eSaral in which trigonometric functions describe the relationship between the angle of the right-angled triangle and the ratio of the length of the side of the triangle. Ex 3.3 class 11 maths ch 3 has provided a deep explanation of topics to prepare for exams. Students can also download the PDF version of these solutions and score well in their tests and examinations. Download the free PDF from the link given below and practice all the questions of exercise 3.3.

## Topics Covered in Exercise 3.3 Class 11 Mathematics Questions

Ex 3.3 class 11 maths chapter 3 solutions cover essential topics like trigonometric functions of sum and difference of two angles. You will be able to get detailed solutions of the topic provided by subject experts of eSaral.

 1 Trigonometric Functions of Sum and Difference of Two Angles
1. Trigonometric Functions of Sum and Difference of Two Angles

You will learn the sum and difference of two numbers (angles) and associated expressions will be derived from trigonometric functions. The fundamental results of this connection are referred to as trigonometric identities.

We know that: 1. sin (– x) = – sin x

2. cos (– x) = cos x

Let’s look at some of the other identities.

3. cos (x + y) = cos x cos y – sin x sin y

4. cos (x – y) = cos x cos y + sin x sin y

5. $\cos \left(\frac{x}{2}-2\right)=\sin \mathrm{x}$

6. $\sin \left(\frac{x}{\pi}-2\right)=\cos x$

7. sin (x + y) = sin x cos y + cos x sin y

8. sin (x – y) = sin x cos y – cos x sin y

9. The following results can be obtained by substituting the appropriate x and y values in Identity 3, Identity 4, Identity 7, and Identity 8.

$\cos \left(\frac{\pi}{2}+\right) 2=-\sin x$

$\sin \left(\frac{\pi}{2}+\right) 2=\cos x$

cos (π – x) = – cos x

sin (π – x) = sin x

cos (π + x) = – cos x

sin (π + x) = – sin x

cos (2π – x) = cos x

sin (2π – x) = – sin x

Similar results for tan x, cot x, sec x and cosec x can be obtained from the results of sin x and cos x

10. If none of the angles x, y and (x + y) is an odd multiple of , then $\frac{\pi}{2}$, then

$\tan (x+y)=\frac{\tan x+\tan y}{1-\tan x \tan y}$

11. $\tan (x-y)=\frac{\tan x-\tan y}{1+\tan x \tan y}$

12. If none of the angles x, y and (x + y) is a multiple of π, then

$\cot (x+y)=\frac{\cot x \cot y-1}{\cot y+\cot x}$

13. $\cot (\mathrm{x}-\mathrm{y})=\frac{\cot x \cot y+1}{\cot y-\cot x}$

14.$\cos 2 \mathrm{x}=\cos ^2 \mathrm{x}-\sin ^2 \mathrm{x}=2 \cos ^2 \mathrm{x}-1=1-2 \sin ^2 \mathrm{x}=\frac{1-\tan ^2 x}{1+\tan ^2 x}$

15. $\sin 2 x=2 \sin x \cos x=\frac{2 \tan x}{1+\tan ^2 x}$

16. $\tan 2 x=\frac{2 \tan x}{1-\tan ^2 x}$

17. sin 3x = 3 sin x – 4 sin3 x

18. cos 3x= 4 cos3 x – 3 cos x

19. $\tan 3 x=\frac{3 \tan x-\tan ^3 x}{1-3 \tan ^2 x}$;

20. (i) $\cos x+\cos y=2 \cos \frac{x+y}{2} \cos \frac{x-y}{2}$

(ii) $\cos x-\cos y=2 \sin \frac{x+y}{2} \sin \frac{x-y}{2}$

(iii) $\sin x+\sin y=2 \sin \frac{x+y}{2} \cos \frac{x-y}{2}$

(iv) $\sin x-\sin y=2 \cos \frac{x+y}{2} \sin \frac{x-y}{2}$

21. As a part of identities given in 20, we can prove the following results:

(i) 2 cos x cos y = cos (x + y) + cos (x – y)

(ii) –2 sin x sin y = cos (x + y) – cos (x – y)

(iii) 2 sin x cos y = sin (x + y) + sin (x – y)

(iv) 2 cos x sin y = sin (x + y) – sin (x – y).

## Tips for Solving Exercise 3.3 Class 11 Chapter 3 Trigonometric Functions

NCERT solutions for class 11 maths chapter 3 ex 3.3 is designed to encourage logical thinking. The interactive nature of these resources and the tips provided by eSaral experts will help you understand this exercise better.

1. It is recommended that students practice the formulas on a regular basis in order to create a strong base.

2. It is essential for students to understand the subject matter and definitions of this exercise. Understanding the explanations provided in chapter 3 of class 11 maths NCERT solutions is a convenient way to gain a complete comprehension of each concept step by step.

3. All the terms, formulas, and techniques mentioned in these solutions are useful for exam preparation. You must remember this before solving exercise 3.3.

## Importance of Solving Ex 3.3 Class 11 Maths Chapter 3 Trigonometric Functions

Ex 3.3 class 11 maths chapter 3 trigonometric functions has numerous benefits of solving questions. Some of the main benefits are described here. You can read them out to solve questions of ex 3.3.

1. NCERT solutions chapter 3 ex 3.3 includes some important identities which you must learn to solve questions with ease.

2. Ex 3.3 class 11 maths ch 3 describe the key concepts of trigonometric functions of sum and difference of two angles in simple explanation so that you can solve the complex questions of this exercise.

3. By practicing questions and examples of ex 3.3 will boost your confidence.

4. NCERT solutions PDF will give you accurate and simple answers which you can cross check while solving questions.