Question

What is the value of \(\tan \left(\frac{\pi}{12}\right)?\)

• A. \(1 - \sqrt{3}\)
• B. \(\sqrt{3} - 1\)
• C. \(2 - \sqrt{3}\)
• D. \(\sqrt{3} - 2\)

Hint:

Longitudinal mode spacing in a laser cavity depends on the mirror separation.

Answer 1

The Correct Option is C

Use identity: \[ \tan\left(\frac{\pi}{12}\right) = \tan\left(15^\circ\right) = \tan(45^\circ - 30^\circ) \Rightarrow \frac{\tan 45^\circ - \tan 30^\circ}{1 + \tan 45^\circ \tan 30^\circ} = \frac{1 - \frac{1}{\sqrt{3}}}{1 + 1 \cdot \frac{1}{\sqrt{3}}} = \frac{\frac{\sqrt{3} - 1}{\sqrt{3}}}{\frac{\sqrt{3} + 1}{\sqrt{3}}} = \frac{\sqrt{3} - 1}{\sqrt{3} + 1} \] Rationalize: \[ = \frac{(\sqrt{3} - 1)^2}{3 - 1} = \frac{4 - 2\sqrt{3}}{2} = 2 - \sqrt{3} \]