Question

A line makes the same angle $\theta$ with each of the $X$ and $Z$-axes. If the angle $\beta$, which it makes with $Y$-axis, is such that ${\sin }^2\beta =3{\sin }^2\theta$, then ${\cos }^2\theta$ equals

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

Answer 1

The correct option is C $\frac{3}{5}$

Let $l,m$ and $n$ be the direction cosines

Then, $l=\cos \theta ,m=\cos \beta ,n=\cos \theta$

We have $l^2+m^2+n^2=1$

$\Rightarrow {\cos }^2\theta +{\cos }^2\beta +{\cos }^2\theta =1$

$\Rightarrow 2{\cos }^2\theta -{\sin }^2\beta =0$

$\Rightarrow 2{\cos }^2\theta -3{\sin }^2\beta =0[\because {\sin }^2\beta =3{\sin }^2\theta ]$

$\Rightarrow {\tan }^2\theta =\frac{2}{3}$

Therefore, ${\cos }^2\theta =\frac{1}{1+{\tan }^2\theta }=\frac{3}{5}$