Question

If $\theta$ is an angle given by $cos\,\theta = \frac{cos^{2} \,\alpha + cos^{2} \,\beta + cos^{2} \,\gamma }{ sin^{2} \,\alpha + sin^{2} \,\beta + sin^{2} \,\gamma }$
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 $\alpha = \beta = \gamma \Rightarrow cos^{2} \,\alpha = cos^{2} \,\beta = cos^{2} \,\gamma$
$\because cos^{2} \,\alpha + cos^{2} \,\beta + cos^{2} \,\gamma = 1$
$\Rightarrow \quad 3\,cos^{2} \,\alpha = 3\,cos^{2} \,\beta = 3\,cos^{2} \,\gamma = 1$
$\Rightarrow \quad cos^{2} \,\alpha = cos^{2} \,\beta = cos^{2} \,\gamma = \frac{1}{3}$
$\therefore sin^{2} \,\alpha = sin^{2} \,\beta = sin^{2} \,\gamma = \frac{2}{3}$
$\therefore \quad cos \,\theta = \frac{3 \left(1/ 3\right)}{3 \left(2/ 3\right)} = \frac{1}{2} \Rightarrow \theta = \frac{\pi}{3}$