## Trigonometry Questions and Answers Part-6

1. If $\alpha+\beta=\pi/2$   and $\beta+\gamma=\alpha$   , then tan $\alpha$ equals
a) $2\left(\tan\beta+\tan\gamma\right)$
b) $\tan\beta+\tan\gamma$
c) $\tan\beta+2\tan\gamma$
d) $2\tan\beta+\tan\gamma$

Explanation:

2. The number of integral values of k for which the equation 7 cos x + 5 sin x = 2k + 1 has a solution is
a) 4
b) 8
c) 10
d) 12

Explanation: The given equation can be written as

3.If $x_{n+1}=\sqrt{\frac{1}{2}\left(1+x_{n}\right)}$     , then $\cos \left[\frac{\sqrt{1-x_0^2}}{x_{1}x_{2}x_{3}.... infinite }\right]\left(-1 < x_{0} < 1\right)$
is equal to
a) -1
b) 1
c) $x_{0}$
d) $1/x_{0}$

Explanation:

4. The simplest value of $\frac{2}{\sqrt{{2}+\sqrt{{2}+\sqrt{2+2\cos4x}}}}$
is
a) $\sec\left(x/2\right)$
b) $\sec x$
c) cosec x
d) 1

Explanation: Given expression is equal to

5. If $\alpha,\beta$   are positive acute angles and $\cos 2\alpha =\frac{3 \cos 2\beta-1}{3-\cos2\beta}$
then $\tan\alpha=k \tan\beta$    such that
a) $k=-\sqrt{2}$
b) $k=\sqrt{2}$
c) k=1
d) $k=\sqrt{3}$

Explanation:

6. If $\tan\alpha+\cot\alpha =p$     , then
a) $\tan^{2}\alpha+\cot^{2}\alpha =p^{2}-2$
b) $\tan^{3}\alpha+\cot^{3}\alpha =p\left(p^{2}-3\right)$
c) $\tan^{5}\alpha+\cot^{5}\alpha =p^{5}-5p^{3}+5$
d) All of the Above

Explanation:

7. If $m\tan\left(\theta-30^{\circ}\right)=n\tan\left(\theta+120^{\circ}\right)$       , then
a) $\frac{m}{n}=\frac{3-4 \sin^{2}\theta}{1-4 \sin^{2}\theta}$
b) $\cos 2\theta=\frac{m+n}{2\left(m-n\right)}$
c) $\frac{m-n}{m+n}=2\sec 2\theta$
d) Both a and b

Explanation:

8. If $\frac{\tan 3 A}{\tan A}=k$   , then
a) $\tan^{2} A=\frac{k-3}{3k-1}$
b) $\frac{\\cos 3A}{\cos A}=\frac{2}{k-1}$
c) $\frac{\sin 3A}{\sin A}=\frac{2k}{k-1}$
d) All of the above

Explanation:

9. If $\tan\frac{\alpha}{2},\tan\frac{\beta}{2}$     are the roots of the equation $8x^{2}-26x+15=0$    , then
a) $\tan\frac{\alpha+\beta}{2}=\frac{-26}{7}$
b) $\cos\left(\alpha+\beta\right)=\frac{-627}{725}$
c) $\sin\left(\alpha+\beta\right)=\frac{-364}{725}$
d) All of the above

10. $\left(m+2\right)\sin\theta+\left(2m-1\right)\cos\theta=2m+1$       , if
a) $\tan\theta = 3/4$
b) $\tan\theta = 4/3$
c) $\tan\theta = 2m\left(m^{2}-1\right)$