KC Sinha Mathematics Solution Class 11 Chapter 8 Trigonometric Equations Exercise 8.1

 Exercise 8.1 

Page no 8.21

Type 1

Question 1

Find the principal solutions of the following trigonometric equations :

(i) $\sin x=\frac{1}{2}$

(ii) $\tan x=-1$

(iii) $\sin x=-\frac{\sqrt{3}}{2}$

(iv) $\tan x=\frac{1}{\sqrt{3}}$

(v) $\cot x=\frac{1}{\sqrt{3}}$

(vi) $\operatorname{cosec} x=-2$

(vii) $\tan x=-\sqrt{3}$

(viii) $\sec x=-\frac{2}{\sqrt{3}}$

(ix) $\tan x=\sqrt{3}$

(x) $\sin x=\frac{\sqrt{3}}{2}$

(xi) $\tan x=-\frac{1}{\sqrt{3}}$

(xii) $\operatorname{cosec} x=2$

Page no 8.22

Type 2

Find the general solution of the following trigonometric equations :

(i) $\tan 5 x=0$

(ii) $\sin 3 x=0$

(iii) $\sin ^{2} 2 x=0$

(iv) $\cot 4 x=0$

(v) $\cos ^{3} 4 x=0$

(vi) $\cot ^{2} 5 \theta=0$

Question 3

Find the general solution of the following trigonometric equations :

(i) $\sin \theta=\frac{\sqrt{3}}{2}$

(ii) $\tan 3 \theta=-1$

(iii) $\sec 2 \theta=-2$

(iv) $\cos \theta=-\frac{\sqrt{3}}{2}$

(v) $\cos 5 x=-\frac{1}{\sqrt{2}}$

(vi) $\sin 3 x=\frac{1}{\sqrt{2}}$

(vii) $\cos x=\frac{1}{2}$

(viii) $\sin x=-\frac{\sqrt{3}}{2}$

Type 3

Question 4

Solve the following trigonometric equations :

(i) $3 \cos ^{2} \theta+7 \sin ^{2} \theta=4$

(ii) $\tan x+\tan 2 x+\tan x \tan 2 x=1$

(iii) $\cos 4 x=\cos 2 x$

(iv) $\tan x+\tan 2 x+\tan 3 x=\tan x \tan 2 x \tan 3 x$

(v) $\sin 3 x+\cos 2 x=0$

(vi) $\tan \left(\frac{\pi}{4}+\theta\right)+\tan \left(\frac{\pi}{4}-\theta\right)=4$

(vii) $\tan 2 x=-\cot \left(x+\frac{\pi}{3}\right)$

Question 5

Solve the following equations :

(i) $\sin a x+\cos b x=0$

(iii) $\cos \theta=\sin 105^{\circ}+\cos 105^{\circ}$

(ii) $\tan x \tan 4 x=1$

Question 6

Solve the equation $\tan x+\cot x=2$

Question 7

Solve the equation $\tan ^{2} x+\cot ^{2} x=2$

Question 8

Solve the equation $\sec ^{2} 2 x=1-\tan 2 x$

Type 4

Question 9

Find the general solution of the following trigonometric equations :

(i) $\sin 2 x+\cos x=0$

(iii) $2 \cos ^{2} x+3 \sin x=0$

(ii) $\tan ^{3} x-3 \tan x=0$

(v) $\sin 2 x+\sin 4 x+\sin 6 x=0$

(iv) $\cos 3 x+\cos x-2 \cos x=0$

(vii) $\sin 2 x-\sin 4 x+\sin 6 x=0$

(vi) $\sin x+\sin 3 x+\sin 5 x=0$

(viii) $\cos 3 x+\cos x-\cos 2 x=0$

Page no 8.23

Type 5

Question 10

(i) $\sqrt{3} \cos x-\sin x=1$

(ii) $\cos \theta-\sin \theta=-1$

(iii) $\cos x+\sin x=1$

(iv) $\sec x-\tan x=\sqrt{3}$

(v) $\sqrt{2} \sec x+\tan x=1$

(vi) $\cot x+\operatorname{cosec} x=\sqrt{3}$

(vii) $\sin x+\cos x=\frac{1}{\sqrt{2}}$

(viii) $\cos x+\sqrt{3} \sin x=1$