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 60°.
∴ Principal value of x = 60° = $\frac{π}{3}$ radians.
If tan x = tan α, when a is the principal value of θ,
then x = nπ + α.
∴ General value of x = nπ + $\frac{π}{3}$

2. sec x = 2 = sec 60° or cos x = $\frac{1}{2}$ = cos 60°.
∴ Principal value = 60°= $\frac{π}{3}$ radians.
For cos θ = cos α, θ = 2nπ ± α.
∴ General value of x = 2nn ± $\frac{π}{3}$.

3. cot x = – $\sqrt{3}$ ⇒ tan x = – $\frac{1}{\sqrt{3}}$
Now, tan 30° = $\frac{1}{\sqrt{3}}$ ⇒ tan (180° – 30°) = – tan 30°.
= – $\frac{1}{\sqrt{3}}$ or tan 150° = – $\frac{1}{\sqrt{3}}$.
Thus, principle value of x = 150° = $\frac{5π}{6}$ radians.
∴ General value of x = nπ + α
= nπ + $\frac{5π}{6}$

4. cosec x = – 2 or sin x = – $\frac{1}{2}$
sin 30° = $\frac{1}{2}$ or sin (- 30°) = – sin 30° = – $\frac{1}{2}$.
∴ Principal value of x = – 30° = – $\frac{π}{6}$.
So, general value of x = nπ + (- 1)ⁿα
= nπ + (- 1)ⁿ (- $\frac{π}{6}$) = nπ – (- 1)ⁿ ($\frac{π}{6}$).

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 = nπ or x = $\frac{nπ}{3}$.
If sin x = 0, then x = nπ.

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)$\frac{π}{2}$ ⇒ x = (2n + 1)$\frac{π}{4}$.
If 2cos x – 1 = 0, cos x = $\frac{1}{2}$ = cos 60° = cos$\frac{π}{3}$.
⇒ x = 2nπ ± $\frac{π}{3}$.

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(π + $\frac{π}{6}$) = sin $\frac{7π}{6}$ ⇒ x = nπ + (- 1)ⁿ $\frac{7π}{6}$.

8. sec² 2x = 1 – tan 2x
⇒ 1 + tan² 2x = 1 – tan 2x = 0
⇒ tan² 2x + tan 2x = 0
⇒ tan 2x(tan 2x + 1) = 0.
If tan 2x = 0, then 2x = nπ or x = $\frac{nπ}{2}$
If tan 2x + 1 = 0, then tan 2x = – 1 = tan (π – $\frac{π}{4}$) = tan $\frac{3π}{4}$
⇒ 2x = nπ + $\frac{3π}{4}$ or x = $\frac{nπ}{2}$ + $\frac{3π}{8}$.

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 = nπ or x = $\frac{nπ}{3}$
If 2cos 2x + 1 = 0, then cos 2x = – $\frac{1}{2}$ = cos (π – $\frac{π}{3}$) = cos $\frac{2π}{3}$
∴ 2x = 2nπ ± $\frac{2π}{3}$ or x = nπ ± $\frac{π}{3}$.