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 ${\sin }^2\beta =3{\sin }_{\Theta }^2$, then ${\cos }^2\Theta$ equals:

• 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
$\therefore l=\cos \theta ,m=\cos \beta ,n=\cos \theta$
we know that $l^2+m^2+n^2=1$
$\Rightarrow {\cos }^2\theta +{\cos }^2\beta +{\cos }^2\theta =1\Rightarrow 2{\cos }^2\theta =1-{\cos }^2\beta$
$\Rightarrow 2{\cos }^2\theta ={\sin }^2\beta$ ...(1)
But ${\sin }^2\beta =3{\sin }^2\theta$ ...(2)
From (1) and (2)
$3{\sin }^2\theta =2{\cos }^2\theta \Rightarrow 3(1-{\cos }^2\theta )=2{\cos }^2\theta \Rightarrow 3-3{\cos }^2\theta =2{\cos }^2\theta \Rightarrow 3=5{\cos }^2\theta$
$\Rightarrow {\cos }^2\theta =\frac{3}{5}$