NCERT Solutions for Class 11 Maths Chapter 3 Trigonometric Functions Exercise 3.2

NCERT Solutions for Class 11 Maths Chapter 3 Exercise 3.2 focus on the concepts of trigonometric identities and their applications. This exercise helps students learn how to use the fundamental identities like Pythagorean identity, sum and difference formulas, and other trigonometric relationships to simplify and solve problems.

The exercise will also help students understand the relationship between different trigonometric functions, and how to prove trigonometric identities step-by-step.

NCERT Solutions for Class 11 Maths Chapter 3 Trigonometric Functions Exercise 3.2

Trigonometric Functions

Medium

Q.

Find the angle in radian through which a pendulum swings if its length is 75 cm and the tip describes an arc of length
(i) 10 cm
(ii) 15 cm
(iii) 21 cm

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Trigonometric Functions

Medium

Q.

Find the values of other five trigonometric functions in Exercises 1 to 5.

$1.\cos \text{x}=-\frac{1}{2}, \text{x} \mathrm{lies} \mathrm{in} \mathrm{third} \mathrm{quadrant}.$

$2.\sin \text{x}=\frac{3}{5}, \text{x} \mathrm{lies} \mathrm{in} \mathrm{second} \mathrm{quadrant}.$

$3.\cot \text{x}=\frac{3}{4}, \text{x} \mathrm{lies} \mathrm{in} \mathrm{third} \mathrm{quadrant}.$

$4.\sec \text{x}=\frac{13}{5}, \text{x} \mathrm{lies} \mathrm{in} \mathrm{fourth} \mathrm{quadrant}.$

$5.\tan \text{x}=-\frac{5}{12}, \text{x} \mathrm{lies} \mathrm{in} \mathrm{second} \mathrm{quadrant}.$

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Trigonometric Functions

Medium

Q.

Find the values of the trigonometric functions in Exercises 6 to 10.

6. sin 765°
7. cosec (– 1410°)
$8. \tan \frac{19\mathrm{\pi }}{3}$

$9. \sin (-\frac{11\mathrm{\pi }}{3})$

$10. \cot (-\frac{15\mathrm{\pi }}{4})$

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Trigonometric Functions

Medium

Q.

$\begin{array}{l}\mathrm{Find} \mathrm{the} \mathrm{general} \mathrm{solution} \mathrm{for} \mathrm{each} \mathrm{of} \mathrm{the} \mathrm{following}\\ \mathrm{equation} : \sin \mathrm{x} + \sin 3\mathrm{x} + \sin 5\mathrm{x} = 0\end{array}$

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Trigonometric Functions

Medium

Q.

In any triangle ABC, if a = 18, b = 24, c = 30, find

1. cos A, cos B, cos C
2. sin A, sin B, sin C

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Trigonometric Functions

Medium

Q.

For any triangle ABC, prove that a (b cos C – c cos B) = b² – c²

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Trigonometric Functions

Difficult

Q.

$\begin{array}{l}\text{For any triangle ABC},\text{ prove that}\\ \mathrm{a}(\cos \mathrm{C}-\cos \mathrm{B})=2(\mathrm{b}-\mathrm{c}){\cos }^2\frac{\mathrm{A}}{2}\end{array}$

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Trigonometric Functions

Difficult

Q.

$\begin{array}{l}\text{For any triangle ABC},\text{ prove that}\\ \frac{\sin (\mathrm{B}-\mathrm{C})}{\sin (\mathrm{B}+\mathrm{C})}=\frac{{\mathrm{b}}^2-{\mathrm{c}}^2}{{\mathrm{a}}^2}\end{array}$

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Trigonometric Functions

Difficult

Q.

$\begin{array}{l}\text{For any triangle ABC},\text{ prove that}\\ (\mathrm{b}+\mathrm{c}) \cos \frac{\mathrm{B} + \mathrm{C}}{2} = \mathrm{a} \cos \frac{\mathrm{B}-\mathrm{C}}{2}\end{array}$

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Question 1: Find the values of the other five trigonometric functions if cos x = -1/2, x lies in the third quadrant.

• sec x: cos x ka ulta hota hai, isliye sec x = 1 / (-1/2) = -2.
• sin x: Formula sin^2 x + cos^2 x = 1 se, sin^2 x = 1 - (-1/2)^2 = 3/4. Third quadrant mein sin negative hota hai, toh sin x = -sqrt(3) / 2.
• cosec x: sin x ka ulta, yani cosec x = -2 / sqrt(3).
• tan x: sin x / cos x = (-sqrt(3)/2) / (-1/2) = sqrt(3).
• cot x: tan x ka ulta, yani cot x = 1 / sqrt(3).

Question 2: Find the values of other five trigonometric functions if sin x = 3/5, x lies in second quadrant.

• cosec x: sin x ka ulta, isliye cosec x = 5/3.
• cos x: cos^2 x = 1 - sin^2 x = 1 - (3/5)^2 = 16/25. Second quadrant mein cos negative hota hai, toh cos x = -4/5.
• sec x: cos x ka ulta, yani sec x = -5/4.
• tan x: sin x / cos x = (3/5) / (-4/5) = -3/4.
• cot x: tan x ka ulta, yani cot x = -4/3.

Question 3: Find the values of other five trigonometric functions if cot x = 3/4, x lies in third quadrant.

• tan x: cot x ka ulta, isliye tan x = 4/3.
• sec x: sec^2 x = 1 + tan^2 x = 1 + (4/3)^2 = 25/9. Third quadrant mein sec negative hota hai, toh sec x = -5/3.
• cos x: sec x ka ulta, yani cos x = -3/5.
• sin x: tan x * cos x = (4/3) * (-3/5) = -4/5.
• cosec x: sin x ka ulta, yani cosec x = -5/4.

Question 4: Find the values of other five trigonometric functions if sec x = 13/5, x lies in fourth quadrant.

• cos x: sec x ka ulta, isliye cos x = 5/13.
• tan x: tan^2 x = sec^2 x - 1 = (13/5)^2 - 1 = 144/25. Fourth quadrant mein tan negative hota hai, toh tan x = -12/5.
• cot x: tan x ka ulta, yani cot x = -5/12.
• sin x: tan x * cos x = (-12/5) * (5/13) = -12/13.
• cosec x: sin x ka ulta, yani cosec x = -13/12.

Question 5: Find the values of other five trigonometric functions if tan x = -5/12, x lies in second quadrant.

• cot x: tan x ka ulta, isliye cot x = -12/5.
• sec x: sec^2 x = 1 + tan^2 x = 1 + (-5/12)^2 = 169/144. Second quadrant mein sec negative hota hai, toh sec x = -13/12.
• cos x: sec x ka ulta, yani cos x = -12/13.
• sin x: tan x * cos x = (-5/12) * (-12/13) = 5/13.
• cosec x: sin x ka ulta, yani cosec x = 13/5.

Question 6 to 10: Find the values of Trigonometric Functions

• 6. sin 765°: sin(2 * 360° + 45°) = sin 45° = 1 / sqrt(2).
• 7. cosec(-1410°): cosec(-1410° + 4 * 360°) = cosec 30° = 2.
• 8. tan 19pi / 3: tan(6pi + pi/3) = tan pi/3 = sqrt(3).
• 9. sin(-11pi / 3): sin(-11pi/3 + 4pi) = sin pi/3 = sqrt(3) / 2.
• 10. cot(-15pi / 4): cot(-15pi/4 + 4pi) = cot pi/4 = 1.

FAQs – Class 11 Maths Chapter 3 Exercise 3.2 Trigonometric Functions

Q1. What is the focus of Exercise 3.2?
Exercise 3.2 focuses on applying and proving trigonometric identities and simplifying trigonometric expressions using various fundamental identities. It also involves solving problems using these identities.

Q2. What are the fundamental trigonometric identities covered in this chapter?
Some of the fundamental trigonometric identities are:

• Pythagorean identities:

   

  $$\sin^2\theta + \cos^2\theta = 1$$sin2θ+cos2θ=1

  $$1 + \tan^2\theta = \sec^2\theta$$1+tan2θ=sec2θ

  $$1 + \cot^2\theta = \csc^2\theta$$1+cot2θ=csc2θ

• Reciprocal identities:

   

  $$\sec\theta = \frac{1}{\cos\theta}, \quad \csc\theta = \frac{1}{\sin\theta}, \quad \cot\theta = \frac{1}{\tan\theta}$$secθ=cosθ1​,cscθ=sinθ1​,cotθ=tanθ1​

Q3. Why is Exercise 3.2 important for board exams?
Exercise 3.2 is important because understanding and applying trigonometric identities is crucial for solving a wide variety of problems in trigonometry. This chapter lays the groundwork for more advanced topics in trigonometry, calculus, and physics.

Q4. How can students prepare effectively for Exercise 3.2?
Students should:

• Thoroughly memorize the trigonometric identities and understand how to apply them.

• Practice proving trigonometric identities and simplifying expressions.

• Work on problems involving substitutions and manipulations using identities.

Q5. What is the significance of proving trigonometric identities?
Proving trigonometric identities is crucial because it helps students understand the fundamental relationships between trigonometric functions and their manipulations. It also strengthens logical reasoning and problem-solving skills.