## NCERT Solutions for Class 11 Maths Chapter 3 Trigonometric Functions Exercise 3.4

NCERT Solutions of Chapter 3 Trigonometric Functions Exercise 3.4 will help you a lot if you want to excel in the examinations. The NCERT Solutions for Class 11 Maths prepared by Studyrankers experts are detailed and accurate so you can clear your doubts easily. You must try to understand the basic facts and important points before trying for advance problems and these NCERT Solutions will help you in developing these skillset.

1. (For Qs. 1-4): Find the principal and general solutions of the following equations:

tanx = √3

**Answer**

tan x = √3 = tan 60 °

∴ Principal value of x = 60° = Ï€/3 radians and

= 4Ï€/3

If tanÎ¸ = tan Î± where ‘Î±’ is the principal value of Î¸.

Then Î¸ = nÏ€+ Î±

∴ General value of x = nÏ€ + Ï€/3, n ∈ z.

2. sec x = 2

**Answer**

sec x = 2 = sec 60 or cos x = 1/2= cos 60

∴ Principal value = 60 = Ï€/3 radians.

For cosÎ¸ = cosÎ± , Î¸ = 2nÏ€ ± Î±

General value of x = 2nÏ€ ± Ï€/3

3. cot x = -√3

**Answer**

cotx = -√3, tan x = - 1/√3

= tan(180 - 30) = tan(Ï€ - Ï€/6) = tan 5Ï€/6

=> x = 5Ï€/6

Principal solution = 5Ï€/6, 11Ï€/6

Genral solution = nÏ€ ± 5Ï€/6, n ∈ z.

4. cosec x = -2

**Answer**

cosec Î¸ = -2 or sin Î¸ = -1/2

sin30 = 1/2 or sin(-30) = -sin30 = -1/2

Principal value of x = -30 = - Ï€/6

general value of x = nÏ€ +(-1)

^{n}Î±
= nÏ€+(-1)

5. Find the general solution of the following equations:

cos 4x = cos 2x

cos 4x = cos 2x

4x = 2nÏ€ ± 2x

Taking + ve sign, we get

4x = 2nÏ€ + 2x

=> 4x + 2x = 2nÏ€

=> 6x = 2nÏ€

x = nÏ€/3

=> x = nÏ€, n ∈ z

Taking -ve sign

4x = 2nÏ€ - 2x

=> 3x + 2x = 2nÏ€

=> 6x = 2nÏ€

=> x = nÏ€/3, n ∈ z

General solution is x = nÏ€/3 or x = nÏ€, n ∈ z

6. Find the general solution of the following equations :

Cos3x + cosx – cos2x = 0

Cos3x + cosx – cos2x = 0

Or 2cos (3x + x)/2 cos(3x – x)/2 – cos2x = 0

Or 2cos2xcosx – cos2x = 0

Or cos2x(2cosx – 1)= 0

If cos 2x = 0,

2x = (2n + 1) Ï€/2 => x = (2n + 1) Ï€/4

If 2cos x – 1 = 0, cos x = 1/2 => cosx = cos Ï€/3

=> x = 2nÏ€ ± Ï€/3

7. Find the general solution of the following equations:

sin2x + cosx = 0

sin 2x + cos x = 0

=> 2cos x cos x cos x = 0

=> cos x (2 sin x + 1) = 0

=> cos x = 0

Or 2 sin x + 1 = 0

=> cos x = 1

Or sin x = -1/2 => cos x = 0

or sin x = sin( Ï€ + Ï€/6) => cos x = 0

Or sin x = sin 7Ï€/6 => x = (2n + 1)Ï€/2

Or x = nÏ€ +(-1)

Hence, general solution is

X = (2n + 1) Ï€/2 or x = nÏ€ +(-1)

where n ∈ Z

8. Find the general solution of

Sec

Sec

=> 1 + tan

=> tan

=> tan 2x(tan 2x + 1) = 0

If tan 2x = 0, 2x = nÏ€ or x = nÏ€/2

If tan 2x + 1 = 0, tan 2x = -1

= tan(Ï€ - Ï€/4) = tan 3Ï€/4

=> 2x = nÏ€ + 3Ï€/4 or x = nÏ€/2 + 3Ï€/8

9. Find the general solution of

sin x + sin3x + sin 5x = 0

We have, (sin 5x + sin x) + sin3x = 0

=> 2sin ((5x + x)/2) cos((5x - x)/2) + sin3x = 0

or 2sin3xcos2x + sin3x = 0

or sin 3x(2cos 2x + 1) = 0

if sin 3x = 0 => 3x = nÏ€ or x = nÏ€/3

if 2cos2x + 1 = 0,

cos2x = -1/2 = cos(Ï€ - Ï€/3) = cos 2Ï€/3

∴ 2x = 2nÏ€ ± 2Ï€/3 or x = nÏ€ ± Ï€/3

^{n}(- Ï€/6) = nÏ€ -(-1)^{2}(Ï€/6)5. Find the general solution of the following equations:

cos 4x = cos 2x

**Answer**cos 4x = cos 2x

4x = 2nÏ€ ± 2x

Taking + ve sign, we get

4x = 2nÏ€ + 2x

=> 4x + 2x = 2nÏ€

=> 6x = 2nÏ€

x = nÏ€/3

=> x = nÏ€, n ∈ z

Taking -ve sign

4x = 2nÏ€ - 2x

=> 3x + 2x = 2nÏ€

=> 6x = 2nÏ€

=> x = nÏ€/3, n ∈ z

General solution is x = nÏ€/3 or x = nÏ€, n ∈ z

6. Find the general solution of the following equations :

Cos3x + cosx – cos2x = 0

**Answer**Cos3x + cosx – cos2x = 0

Or 2cos (3x + x)/2 cos(3x – x)/2 – cos2x = 0

Or 2cos2xcosx – cos2x = 0

Or cos2x(2cosx – 1)= 0

If cos 2x = 0,

2x = (2n + 1) Ï€/2 => x = (2n + 1) Ï€/4

If 2cos x – 1 = 0, cos x = 1/2 => cosx = cos Ï€/3

=> x = 2nÏ€ ± Ï€/3

7. Find the general solution of the following equations:

sin2x + cosx = 0

**Answer**sin 2x + cos x = 0

=> 2cos x cos x cos x = 0

=> cos x (2 sin x + 1) = 0

=> cos x = 0

Or 2 sin x + 1 = 0

=> cos x = 1

Or sin x = -1/2 => cos x = 0

or sin x = sin( Ï€ + Ï€/6) => cos x = 0

Or sin x = sin 7Ï€/6 => x = (2n + 1)Ï€/2

Or x = nÏ€ +(-1)

^{n}7Ï€/6, n ∈ ZHence, general solution is

X = (2n + 1) Ï€/2 or x = nÏ€ +(-1)

^{n}7Ï€/6where n ∈ Z

8. Find the general solution of

Sec

^{2}2x = 1 – tan 2x**Answer**Sec

^{2}2x = 1 – tan 2x=> 1 + tan

^{2}2x = 1 – tan 2x = 0=> tan

^{2}2x + tan2x = 0=> tan 2x(tan 2x + 1) = 0

If tan 2x = 0, 2x = nÏ€ or x = nÏ€/2

If tan 2x + 1 = 0, tan 2x = -1

= tan(Ï€ - Ï€/4) = tan 3Ï€/4

=> 2x = nÏ€ + 3Ï€/4 or x = nÏ€/2 + 3Ï€/8

9. Find the general solution of

sin x + sin3x + sin 5x = 0

**Answer**We have, (sin 5x + sin x) + sin3x = 0

=> 2sin ((5x + x)/2) cos((5x - x)/2) + sin3x = 0

or 2sin3xcos2x + sin3x = 0

or sin 3x(2cos 2x + 1) = 0

if sin 3x = 0 => 3x = nÏ€ or x = nÏ€/3

if 2cos2x + 1 = 0,

cos2x = -1/2 = cos(Ï€ - Ï€/3) = cos 2Ï€/3

∴ 2x = 2nÏ€ ± 2Ï€/3 or x = nÏ€ ± Ï€/3