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 equal to

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

Answer 1

Answer: (C)

$\frac{3}{5}$

A line makes angle $\theta$ with $x$-axis and $z$-axis and $\beta$ with $y$-axis.
$l=\cos \theta ,m=\cos \beta ,n=\cos \theta$ ....$\text{Since}{l}^2+m^2+n^2=1$
$\Rightarrow {\cos }^2\theta +{\cos }^2\beta +{\cos }^2\theta =1$
$\Rightarrow 2{\cos }^2\theta =1-{\cos }^2\beta ={\sin }^2\beta$
$\Rightarrow 2{\cos }^2\theta =3{\sin }^2\theta$ .....(given, ${\sin }^2\beta =3{\sin }^2\theta$)
$\Rightarrow 2{\cos }^2\theta =3(1-{\cos }^2\theta )$
$\Rightarrow 5{\cos }^2\theta =3$
$\Rightarrow {\cos }^2\theta =\frac{3}{5}$