Question

A line makes an angle $\theta $ with the $x-$ axis and the $y-$ axis. If it makes an angle $\alpha $ with the $z-$ axis such that $sin^{2} \alpha =3sin^{2} ⁡ \theta ,$ then $cos 2 \theta $ is equal to

• A. $\frac{\sqrt{3}}{2}$
• B. $\frac{3}{4}$
• C. $\frac{1}{5}$
• D. $\frac{1}{2}$

Answer 1

Answer: (C)

Direction cosines of the line are $\left(cos \theta , cos ⁡ \theta , cos ⁡ \alpha \right)$
As we know that $\cos ^{2} \theta+\cos ^{2} \theta+\cos ^{2} \alpha=1$
$\Rightarrow 2 \cos ^{2} \theta=\sin ^{2} \alpha \ldots \ldots$ (i)
Also, $sin^{2} \alpha =3sin^{2} ⁡ \theta $ (Given)
$\Rightarrow 2cos^{2} \theta =3sin^{2} ⁡ \theta \Rightarrow tan^{2} ⁡ \theta =\frac{2}{3}$
$cos 2 \theta =\frac{1 - tan^{2} ⁡ \theta }{1 + tan^{2} ⁡ \theta }=\frac{1 - \frac{2}{3}}{1 + \frac{2}{3}}=\frac{1}{5}$