## Trigonometry Questions and Answers Part-2

1.If $\sin x+\sin^{2}x=1$    then the value of $\cos^{12} x+3\cos^{10} x+3\cos^{8} x+\cos^{6} x-1$
is equal to
a) 0
b) 1
c) -1
d) 2

Explanation:

2. If $\theta$  lies in the first quadarnt and $\cos \theta=8/17$  , then the value of $\cos\left(30^{\circ}+\theta\right)+\cos\left(45^{\circ}-\theta\right)+\cos\left(120^{\circ}-\theta\right)$
a) $\left(\frac{\sqrt{3}-1}{2}+\frac{1}{\sqrt{2}}\right)\frac{23}{17}$
b) $\left(\frac{\sqrt{3}+1}{2}+\frac{1}{\sqrt{2}}\right)\frac{23}{17}$
c) $\left(\frac{\sqrt{3}-1}{2}-\frac{1}{\sqrt{2}}\right)\frac{23}{17}$
d) $\left(\frac{\sqrt{3}+1}{2}-\frac{1}{\sqrt{2}}\right)\frac{23}{17}$

Explanation:

3. Let$-\frac{\pi}{6}<\theta < - \frac{-\pi}{12}$     . Suppose $\alpha_{1}$ and $\beta_{1}$  are the roots of the equation $x^{2}-2x\sec\theta+1=0$     and $\alpha_{2}$ and $\beta_{2}$ are the roots of the equation $x^{2}+2x\tan\theta-1=0$     . If $\alpha_{1}> \beta_{1}$   and $\alpha_{2}> \beta_{2}$   , then $\alpha_{1}+ \beta_{2}$   equals
a) $2\left(\sec\theta-\tan\theta\right)$
b) $2\sec\theta$
c) $-2\tan\theta$
d) 0

Explanation:

4. The value of the determinant $\begin{bmatrix}1 & a & a^{2} \\\cos\left(n-1\right)x & \cos nx & \cos\left(n+1\right)x \\\sin\left(n-1\right)x & \sin nx & \sin\left(n+1\right)x\end{bmatrix}$
(a $\neq$ 1) is zero if
a) $\sin x=0$
b) $\cos x=0$
c) a=0
d) $\cos x=\frac{1+a^{2}}{2a}$

Explanation:

5. The value of $\cos^{4}\left(\frac{\pi}{8}\right)+\cos^{4}\left(\frac{3\pi}{8}\right)+\cos^{4}\left(\frac{5\pi}{8}\right)+\cos^{4}\left(\frac{7\pi}{8}\right)$
is equal to
a) 1/2
b) 3/4
c) 3/2
d) 1

Explanation:

6. An angle $\alpha$ is divided into two parts so that the ratio of the tangents of these parts is $\lambda$ . If the difference between these parts is x then sin x/sin $\alpha$ is equal to
a) $\gamma/\left(\gamma+1\right)$
b) $\left(\gamma-1\right)/\gamma$
c) $\frac{\gamma-1}{\gamma+1}$
d) $\frac{2\gamma}{\gamma+1}$

Explanation:

7. If $\alpha\epsilon\left(0,\pi/2\right)$   , then the expression $\sqrt{x^{2}+x}+\frac{\tan^{2}x}{\sqrt{x^{2}+x}}$
is always greater than or equal to
a) $2\tan \alpha$
b) 2
c) 1
d) $\sec^{2} \alpha$

Explanation: Since A.M ≥ G.M, we get

8. Given $\theta\epsilon\left(0,\pi/4\right)$    and $t_{1}=\left(\tan\theta\right)^{\tan\theta},t_{2}=\left(\tan\theta\right)^{\cot\theta},t_{3}=\left(\cot\theta\right)^{\tan\theta},t_{4}=\left(\cot\theta\right)^{\cot\theta}$
then
a) $t_{1}>t_{2}>t_{3}> t_{4}$
b) $t_{4}>t_{3}>t_{1}> t_{2}$
c) $t_{3}>t_{1}>t_{2}> t_{4}$
d) $t_{2}>t_{3}>t_{1}> t_{4}$

Explanation:

9. If $x=\sin\alpha,y=\sin\beta,z=\sin\left(\alpha+\beta\right)$
then $\cos\left(\alpha+\beta\right)$   =
a) $\frac{x^{2}+y^{2}+z^{2}}{2xy}$
b) $\frac{x^{2}+y^{2}-z^{2}}{xy}$
c) $\frac{z^{2}-x^{2}-y^{2}}{2xy}$
d) $\frac{z^{2}-x^{2}-y^{2}}{xy}$

10. The radius of the circle $2x^{2}+2y^{2}-4x\cos\theta+4y\sin\theta-1-4\cos\theta-\cos2\theta=0$
a) $1-\cos\theta$
b) $1+\cos\theta$
c) $1-\sin\theta$