Exercise 6.1
Page no - 6.18
Type 1
Question 1
Find the value of :
(i) \cos 210^{\circ}
(ii) \sin 15^{\circ}
(iii) \sin 75^{\circ}
(iv) \tan 15^{\circ}
(v) \tan 75^{\circ}
(vi) \tan \left(-330^{\circ}\right)
(vii) \cos 1395^{\circ}
(viii) \tan \left(\frac{11 \pi}{12}\right)
(ix) \tan \left(\frac{13 \pi}{12}\right)
Question 2
Find the value of the following :
(i) \sin 105^{\circ}+\cos 105^{\circ}
(ii) \sin 300^{\circ} \operatorname{cosec} 1050^{\circ}-\tan \left(-120^{\circ}\right)
(iii) \sin 690^{\circ} \cos 930^{\circ}+\tan \left(-765^{\circ}\right) \operatorname{cosec}\left(-1170^{\circ}\right)
(iv) \cot ^{2} \frac{\pi}{6}+\operatorname{cosec} \frac{5 \pi}{6}+3 \tan ^{2} \frac{\pi}{6}
(v) 3 \sin \frac{\pi}{6} \sec \frac{\pi}{3}-4 \sin \frac{5 \pi}{6} \cot \frac{\pi}{4}
(vi) \sin ^{2} \frac{\pi}{6}+\cos ^{2} \frac{\pi}{3}-\tan ^{2} \frac{\pi}{4}
Question 3
Prove that \sin 75^{\circ}=\frac{\sqrt{6}+\sqrt{2}}{4}
Question 4
Find \tan 15^{\circ} and hence show that \tan 15^{\circ}+\cot 15^{\circ}=4
Question 5
Evaluate \tan \left\{(-1)^{n} \frac{\pi}{4}\right\}, where n is an integer.
Type 2
Question 6
If \sin \alpha=\frac{1}{\sqrt{10}}, \sin \beta=\frac{1}{\sqrt{5}} (where \alpha, \beta and \alpha+\beta are positive acute angles). show that \alpha+\beta=\frac{\pi}{4}.
Page no 6.19
Question 7
If \sin A=\frac{3}{5} and \cos B=\frac{9}{41}, 0<A<\frac{\pi}{2}, 0<B<\frac{\pi}{2}. Find the values of the following :
(i) \sin (A+B)
(ii) \cos (A+B)
(iii) \sin (A-B)
(iv) \cos (A-B)
Question 8
If \sin A=\frac{1}{2}, \cos B=\frac{\sqrt{3}}{2}, where \frac{\pi}{2}<A<\pi and 0<B<\frac{\pi}{2}, find the following :
(i) \tan (A+B)
(ii) \tan (A-B)
Question 9
If \sin x=\frac{3}{5}, \cos y=-\frac{12}{13}, where \frac{\pi}{2}<x<\pi and \frac{\pi}{2}<y<\pi, show that \sin (x+y)=-\frac{56}{65}
Question 10
Evaluate :
(i) \cos ^{2}\left(\frac{\pi}{4}+x\right)-\sin ^{2}\left(\frac{\pi}{4}-x\right)
(ii) \sin ^{2}\left(15^{\circ}+A\right)-\sin ^{2}\left(15^{\circ}-A\right)
(iii) \sin ^{2}\left(\frac{\pi}{8}+\frac{A}{2}\right)-\sin ^{2}\left(\frac{\pi}{8}-\frac{A}{2}\right)
Question 11
Prove that
(i) \cos \left(\frac{3 \pi}{2}+x\right) \cos (2 \pi+x)\left[\cot \left(\frac{3 \pi}{2}-x\right)+\cot (2 \pi+x)\right]=1
(ii) \frac{\cos (\pi+\theta) \cdot \cos (-\theta)}{\sin (\pi-\theta) \cdot \cos \left(\frac{\pi}{2}+\theta\right)}=\cot ^{2} \theta
Question 12
Show that
(i) \cos 70^{\circ}, \cos 10^{\circ}+\sin 70^{\circ} \cdot \sin 10^{\circ}=\frac{1}{2}
(ii) \sin 78^{\circ} \cdot \cos 18^{\circ}-\cos 78^{\circ} \cdot \sin 18^{\circ}=\frac{\sqrt{3}}{2}
(iii) \sin \left(40^{\circ}+\theta\right) \cdot \cos \left(10^{\circ}+\theta\right)-\cos \left(40^{\circ}+\theta\right) \cdot \sin \left(10^{\circ}+\theta\right)=\frac{1}{2}
(iv) \sin (n+1) x \sin (n+2) x+\cos (n+1) x \cos (n+2) x=\cos x
(v) \cos \left(\frac{\pi}{4}-x\right) \cos \left(\frac{\pi}{4}-y\right)-\sin \left(\frac{\pi}{4}-x\right) \sin \left(\frac{\pi}{4}-y\right)=\sin (x+y)
(vi) (\sin 3 x+\sin x) \sin x+(\cos 3 x-\cos x) \cos x=0
(vii) \frac{\tan \left(\frac{\pi}{4}+x\right)}{\tan \left(\frac{\pi}{4}-x\right)}=\left(\frac{1+\tan x}{1-\tan x}\right)^{2}
Question 13
Prove that \frac{\cos 20^{\circ}-\sin 20^{\circ}}{\cos 20^{\circ}+\sin 20^{\circ}}=\tan 25^{\circ}.
Page no - 6.20
Question 14
Prove that \cos 9^{\circ}+\cos 9^{\circ}=\sqrt{2} \sin 54^{\circ}
Question 15
Prove that
\frac{\tan A+\tan B}{\tan A-\tan B}=\frac{\sin (A+B)}{\sin (A-B)}
Question 16
Prove that
(i) \tan 8 \theta-\tan 6 \theta-\tan 2 \theta=\tan 8 \theta \tan 6 \theta \tan 2 \theta
(ii) \tan 9^{\circ}+\tan 36^{\circ}+\tan 9^{\circ} \tan 36^{\circ}=1
(iii) \tan 3 x-\tan 2 x-\tan x=\tan 3 x \tan 2 x \tan x
Question 17
Show that
(i) \frac{1}{\tan 3 A-\tan A}-\frac{1}{\cot 3 A-\cot A}=\cot 2 A
(ii) \frac{1}{\tan 3 A+\tan A}-\frac{1}{\cot 3 A+\cot A}=\cot 4 A
Question 18
Prove that
(i) \sin ^{2} 6 x-\sin ^{2} 4 x=\sin 2 x \sin 10 x
(ii) \tan (\alpha+\beta) \tan (\alpha-\beta)=\frac{\sin ^{2} \alpha-\sin ^{2} \beta}{\cos ^{2} \alpha-\sin ^{2} \beta}
Question 19
Prove that :
(i) \sin (A+B) \sin (A-B)+\sin (B+C) \sin (B-C)+\sin (C+A) \sin (C-A)=0
(ii) \tan \{(2 n+1) \pi+\theta\}+\tan \{(2 n+1) \pi-\theta \mid=0\}
Type 3
Question 20
If \tan \alpha=\frac{m}{m+1}, \tan \beta=\frac{1}{2 m+1}, prove that \alpha+\beta=\frac{\pi}{4}
Question 21
If A+B=\frac{\pi}{4}, show that (\cot A-1)(\cot B-1)=2.
Question 22
If 8 \theta=\pi, show that \cos 7 \theta+\cot \theta=0.
Question 23
If \tan \alpha-\tan \beta=x and \cot \beta-\cot \alpha=y, prove that \cot (\alpha-\beta)=\frac{x+y}{x y}
Question 24
If \tan \alpha=2 \tan \beta, show that \frac{\sin (\alpha+\beta)}{\sin (\alpha-\beta)}=3.
Question 25
If \cos A+\sin B=m and \sin A+\cos B=n. prove that 2 \sin (A+B)=m^{2}+n^{2}-2.
Question 26
If \tan x+\tan \left(\frac{\pi}{3}+x\right)+\tan \left(\frac{2 \pi}{3}+x\right)=3, then prove that
Question 27
If a right angle be divided into three parts \alpha, \beta and \gamma, prove that
\cot \alpha=\frac{\tan \beta+\tan \gamma}{1-\tan \beta \tan \gamma} \text {. }
Question 28
If \sin \alpha, \sin \beta-\cos \alpha, \cos \beta=1, show that \tan \alpha+\tan \beta=0.
Page no - 6.21
Question 29
If \sin (\alpha+\beta)=1 and \sin (\alpha-\beta)=\frac{1}{2}, where 0 \leq \alpha, \beta \leq \frac{\pi}{2}, then find the values of \tan (\alpha+2 \beta) and \tan (2 \alpha+\beta)
Question 30
If m \tan \left(\theta-30^{\circ}\right)=n \tan \left(\theta+120^{\circ}\right). show that \cos 2 \theta=\frac{m+n}{2(m-n)}.
Question 31
If \alpha+\beta=\theta and \frac{\tan \alpha}{\tan \beta}=\frac{x}{y}, prove that
\sin (\alpha-\beta)=\frac{x-y}{x+y} \sin \theta
Question 32
If \alpha and \beta are the solutions of the equation a \cos \theta+b \sin \theta=c, then show that
(i) \cos (\alpha+\beta)=\frac{a^{2}-b^{2}}{a^{2}+b^{2}}
(ii) \cos (\alpha-\beta)=\frac{2 c^{2}-\left(a^{2}+b^{2}\right)}{a^{2}+b^{2}}
Type 3
Question 33
Find the maximum and minimum values of the following expressions :
(i) a \cos \theta-b \sin \theta
(ii) 7 \cos \theta+24 \sin \theta
Question 34
Show that \sin 100^{\circ}-\sin 10^{\circ} is positive.
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