## Trigonometry Questions and Answers Part-3

1. If $\tan x+\tan\left(x+\pi/3\right)+\tan\left(x+2\pi/3\right)=3$       , then
a) $\tan x=1$
b) $\tan 2x=1$
c) $\tan 3x=1$
d) none of these.

Explanation:

2. The equation cos $2x+a\sin x=2a-7$     possesses a solution if
a) a < 2
b) $2 \leq a\leq6$
c) a> 6
d) a is any integer

Explanation: The given equation can be written as

3. $\sin47^{\circ}+\sin61^{\circ}-\sin11^{\circ}-\sin25^{\circ}$       is equal to
a) $\sin36^{\circ}$
b) $\cos36^{\circ}$
c) $\sin7^{\circ}$
d) $\cos7^{\circ}$

Explanation: The given expression is equal to

4.If tan $\alpha=1/7$   and sin $\beta=1/\sqrt{10}$   where $0<\alpha,\beta<\pi/2$    , then $2\beta$ is equal to
a) $\pi/4-\alpha$
b) $3\pi/4-\alpha$
c) $\pi/8-\alpha/2$
d) $3\pi/8-\alpha/2$

Explanation:

5. If $3\pi/4<\alpha<\pi$   , then $\sqrt{2 \cot \alpha+\frac{1}{\sin^{2}\alpha}}$
is equal to
a) $1+\cot \alpha$
b) $-1-\cot \alpha$
c) $1-\cot \alpha$
d) $-1+\cot \alpha$

Explanation:

6. If $\cos\left(\theta-\alpha\right)=a$    and $\sin\left(\theta-\beta\right)=b\left(0<\theta-\alpha,\theta-\beta<\pi/2\right)$       , then $\cos^{2}\left(\alpha-\beta\right)+2b\sin\left(\alpha-\beta\right)$
is equal to
a) $4a^{2}b^{2}$
b) $a^{2}-b^{2}$
c) $a^{2}+b^{2}$
d) $-a^{2}b^{2}$

Explanation:

7. The expression $\frac{\cos 6x+6\cos 4x+15\cos 2x+10}{\cos 5x+5\cos 3x+10\cos x}$
is equal to
a) $\cos 2x$
b) $2\cos x$
c) $\cos^{2}x$
d) $1+\cos x$

Explanation: The given expression can be written as

8. If $\tan\left(\pi\cos\theta\right)=\cot\left(\pi\sin\theta\right)$      then $\cos\left(\theta-\pi/4\right)$    is equal to
a) $\pm\frac{1}{2\sqrt{2}}$
b) $\pm\frac{1}{\sqrt{2}}$
c) $\pm\sqrt{2}$
d) $\pm2\sqrt{2}$

Explanation:

9. If $\tan\theta_{1},\tan\theta_{2},\tan\theta_{3}\tan\theta_{4}$      are the roots of the equation $x^{4}-x^{3}\sin2\beta+x^{2}\cos2\beta-x\cos\beta-\sin\beta=0$
then $\tan\left(\theta_{1}+\theta_{2}+\theta_{3}+\theta_{4}\right)$     is equal to
a) $\sin\beta$
b) $\cos\beta$
c) $\tan\beta$
d) $\cot\beta$

Explanation: From the given equation we get

10. The expression $\cos^{2}\phi+\cos^{2}\left( a+\phi\right)-2cos a\cos\phi\cos\left( a+\phi\right)$
is independent of
a) $\phi$
b) a
c) both a and $\phi$
d) none of a and $\phi$