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 ${\cos }^2\theta$ is-

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

Answer 1

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

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

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

$\Rightarrow 1-3{\sin }^2\theta =2{\sin }^2\theta -1\Rightarrow {\sin }^2\theta =\frac{2}{5}$

$\Rightarrow {\cos }^2\theta =\frac{3}{5}$