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 $si{n}^2\alpha =3si{n}^2\theta ,$ then $cos2\theta$ 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 ${\cos }^2\theta +{\cos }^2\theta +{\cos }^2\alpha =1$
$\Rightarrow 2{\cos }^2\theta ={\sin }^2\alpha \dots \dots$ (i)
Also, $si{n}^2\alpha =3si{n}^2\theta$ (Given)
$\Rightarrow 2co{s}^2\theta =3si{n}^2\theta \Rightarrow ta{n}^2\theta =\frac{2}{3}$
$cos2\theta =\frac{1-ta{n}^2\theta }{1+ta{n}^2\theta }=\frac{1-\frac{2}{3}}{1+\frac{2}{3}}=\frac{1}{5}$