# GSEB Solutions Class 11 Maths Chapter 3 Trigonometric Functions Ex 3.4

Gujarat Board GSEB Textbook Solutions Class 11 Maths Chapter 3 Trigonometric Functions Ex 3.4 Textbook Questions and Answers.

## Gujarat Board Textbook Solutions Class 11 Maths Chapter 3 Trigonometric Functions Ex 3.4

Find the principal and general solutions of the following equations:
1. tan x = $$\sqrt{3}$$
2. sec x = 2
3. cot x = – $$\sqrt{3}$$
4. cosec x = – 2
Solutions to questions 1 – 4:
1. tan x = $$\sqrt{3}$$ = tan 60Â°.
âˆ´ Principal value of x = 60Â° = $$\frac{Ï€}{3}$$ radians.
If tan x = tan Î±, when a is the principal value of Î¸,
then x = nÏ€ + Î±.
âˆ´ General value of x = nÏ€ + $$\frac{Ï€}{3}$$

2. sec x = 2 = sec 60Â° or cos x = $$\frac{1}{2}$$ = cos 60Â°.
âˆ´ Principal value = 60Â°= $$\frac{Ï€}{3}$$ radians.
For cos Î¸ = cos Î±, Î¸ = 2nÏ€ Â± Î±.
âˆ´ General value of x = 2nn Â± $$\frac{Ï€}{3}$$.

3. cot x = – $$\sqrt{3}$$ â‡’ tan x = – $$\frac{1}{\sqrt{3}}$$
Now, tan 30Â° = $$\frac{1}{\sqrt{3}}$$ â‡’ tan (180Â° – 30Â°) = – tan 30Â°.
= – $$\frac{1}{\sqrt{3}}$$ or tan 150Â° = – $$\frac{1}{\sqrt{3}}$$.
Thus, principle value of x = 150Â° = $$\frac{5Ï€}{6}$$ radians.
âˆ´ General value of x = nÏ€ + Î±
= nÏ€ + $$\frac{5Ï€}{6}$$

4. cosec x = – 2 or sin x = – $$\frac{1}{2}$$
sin 30Â° = $$\frac{1}{2}$$ or sin (- 30Â°) = – sin 30Â° = – $$\frac{1}{2}$$.
âˆ´ Principal value of x = – 30Â° = – $$\frac{Ï€}{6}$$.
So, general value of x = nÏ€ + (- 1)nÎ±
= nÏ€ + (- 1)n (- $$\frac{Ï€}{6}$$) = nÏ€ – (- 1)n ($$\frac{Ï€}{6}$$).

Find the solution for each of the following equations:
5. cos 4x = cos 2x
6. cos 3x + cos x – cos 2x = 0
7. sin 2x + cos x = 0
8. sec2 2x = 1 – tan 2x
9. sin x + sin 3x + sin 5x = 0
Solutions to questions 5 – 10:

5. cos 4x = cos 2x
or cos 2x – cos 4x = 0.
or 2 sin $$\frac{2x+4x}{2}$$sin $$\frac{4x-2x}{2}$$ = 0
or 2sin 3x sin x = 0.
If sin 3x = 0, then 3x = nÏ€ or x = $$\frac{nÏ€}{3}$$.
If sin x = 0, then x = nÏ€.

6. cos 3x + cos x – cos 2x = 0
or 2cos $$\frac{3x+x}{2}$$cos $$\frac{3x-x}{2}$$ – cos 2x = 0.
or 2 cos 2x cosx – cos 2x = 0
or cos 2x(2 cos x – 1) = 0.
If cos 2x = 0, then 2x = (2n + 1)$$\frac{Ï€}{2}$$ â‡’ x = (2n + 1)$$\frac{Ï€}{4}$$.
If 2cos x – 1 = 0, cos x = $$\frac{1}{2}$$ = cos 60Â° = cos$$\frac{Ï€}{3}$$.
â‡’ x = 2nÏ€ Â± $$\frac{Ï€}{3}$$.

7. sin 2x + cos x = 0
or 2sin x cos x + cos x = 0
or cos x(2sin x + 1) = 0.
sin x = – $$\frac{1}{2}$$ = sin(Ï€ + $$\frac{Ï€}{6}$$) = sin $$\frac{7Ï€}{6}$$ â‡’ x = nÏ€ + (- 1)n $$\frac{7Ï€}{6}$$.

8. sec2 2x = 1 – tan 2x
â‡’ 1 + tan2 2x = 1 – tan 2x = 0
â‡’ tan2 2x + tan 2x = 0
â‡’ tan 2x(tan 2x + 1) = 0.
If tan 2x = 0, then 2x = nÏ€ or x = $$\frac{nÏ€}{2}$$
If tan 2x + 1 = 0, then tan 2x = – 1 = tan (Ï€ – $$\frac{Ï€}{4}$$) = tan $$\frac{3Ï€}{4}$$
â‡’ 2x = nÏ€ + $$\frac{3Ï€}{4}$$ or x = $$\frac{nÏ€}{2}$$ + $$\frac{3Ï€}{8}$$.

9. sin x + sin 3x + sin 5x = 0
or (sin 5x + sin x) + sin 3x = 0.
or 2 sin $$\frac{5x+x}{2}$$cos $$\frac{5x-x}{2}$$ + sin 3x = 0
or 2 sin 3x cos 2x + sin 3x = 0
or sin 3x(2 cos 2x + 1) = 0.
If sin 3x = 0, then 3x = nÏ€ or x = $$\frac{nÏ€}{3}$$
If 2cos 2x + 1 = 0, then cos 2x = – $$\frac{1}{2}$$ = cos (Ï€ – $$\frac{Ï€}{3}$$) = cos $$\frac{2Ï€}{3}$$
âˆ´ 2x = 2nÏ€ Â± $$\frac{2Ï€}{3}$$ or x = nÏ€ Â± $$\frac{Ï€}{3}$$.