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

Question

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

Dr. Abubacker · Accepted Answer

(c)
Here $\sin^{2}\beta = 3\sin^{2}	heta$ (i)
By the question, $\cos^{2}	heta + \cos^{2}	heta + \cos^{2}\beta = 1$ (ii)
$\Rightarrow \ \cos^{2}\beta = 1 - 2\cos^{2}	heta$ (iii)
Adding (i) and (iii), we get
$1 = 1 + 3\sin^{2}	heta - 2\cos^{2}	heta$
$\Rightarrow 1 = 1 + 3\left( 1 - \cos^{2}	heta ight) - 2\cos^{2}	heta$
$\Rightarrow 5\cos^{2}	heta = 3$
$\Rightarrow \cos^{2}	heta = \frac{3}{5}$

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

Explanation

Reviewed by Dr. Abubacker

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