Question

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

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

Answer 1

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

Given that ${\sin }^2\beta =3{\sin }^2\theta$ ...$\left(1\right)$
Let $\theta$, $\theta$ and $\beta$ be the angle made by the line with co-ordinate axes.
Therefore, ${\cos }^2\theta +{\cos }^2\theta +{\cos }^2\beta =1$
$\Rightarrow 2{\cos }^2\theta ={\sin }^2\beta =3{\sin }^2\theta =3-3{\cos }^2\theta$ from $\left(1\right)$
Therefore, ${\cos }^2\theta =\frac{3}{5}$
Ans: C