Question

A line makes the same angle $\theta$, with each of the $x$ and $z$ axis. 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. $\frac{3}{5}$
• B. $\frac{1}{5}$
• C. $\frac{2}{3}$
• D. $\frac{2}{5}$

Answer 1

Answer: (A)

$\frac{3}{5}$

Direction cosines of line are $\cos \theta ,\cos \beta ,\cos \theta ,$
$\therefore {\cos }^2\theta +{\cos }^2\beta +{\cos }^2\theta =1\Rightarrow 2{\cos }^2\theta -{\sin }^2\beta =0$
$\Rightarrow 2{\cos }^2\theta -3{\sin }^2\theta =0$
On substituting ${\sin }^2\theta =1-{\cos }^2\theta$, we get

$2{\cos }^2\theta -3(1-{\cos }^2\theta )=0$
$\Rightarrow 5{\cos }^2\theta =3\Rightarrow {\cos }^2\theta =\frac{3}{5}$