Question

If $\theta$ is an angle given by $cos\theta =\frac{co{s}^2\alpha +co{s}^2\beta +co{s}^2\gamma }{si{n}^2\alpha +si{n}^2\beta +si{n}^2\gamma }$
where $\alpha ,\beta ,\gamma$ are the equal angles made by a line with the positive directions of the axes, then the measure of $\theta$ is

• A. $\frac{\pi }{3}$
• B. $\frac{\pi }{6}$
• C. $\frac{\pi }{2}$
• D. $\frac{\pi }{4}$

Answer 1

Answer: (A)

Since $\alpha =\beta =\gamma \Rightarrow co{s}^2\alpha =co{s}^2\beta =co{s}^2\gamma$
$\because co{s}^2\alpha +co{s}^2\beta +co{s}^2\gamma =1$
$\Rightarrow 3co{s}^2\alpha =3co{s}^2\beta =3co{s}^2\gamma =1$
$\Rightarrow co{s}^2\alpha =co{s}^2\beta =co{s}^2\gamma =\frac{1}{3}$
$\therefore si{n}^2\alpha =si{n}^2\beta =si{n}^2\gamma =\frac{2}{3}$
$\therefore cos\theta =\frac{3\left(1/3\right)}{3\left(2/3\right)}=\frac{1}{2}\Rightarrow \theta =\frac{\pi }{3}$