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)

$\frac{3}{5}$

If a line makes the angle $\alpha ,\beta ,\gamma$ with x,y,z axix respectively then
$l^2+m^2+n^2=1$
$\Rightarrow 2l^2+m^2=1or2n^2+m^2=1$
$\Rightarrow 2{\cos }^2\theta =1-{\cos }^2\beta (\alpha =\gamma =\theta )$
$2{\cos }^2\theta ={\sin }^2\beta$
$\Rightarrow 2{\cos }^2\theta =3{\sin }^2\theta$
$\Rightarrow 5{\cos }^2\theta =3$