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

Answer 1

Answer: (C)

The direction cosines of the line are $\cos \theta , \cos \beta, \cos\theta$
$ \therefore \cos^{2} \theta + \cos^{2} \beta+ \cos^{2} \theta =1$
$ \Rightarrow 2 \cos^{2} \theta = \sin^{2 } \beta = 3 \sin^{2} \theta$ (given)
$ \Rightarrow 2 \cos^{2} \theta = 3-3 \cos^{2} \theta $
$\therefore \cos^{2} \theta = \frac{ 3}{5}$