Question

Find the indicated value. Use either half angle formulas or sum and difference formulas.

Answer 1

By the angle difference formula for sines, we have:

$\sin 15^\circ = \sin 45^\circ \cos 30^\circ - \sin 30^\circ \cos 45^\circ = \frac{sqrt{6}}{4} - \frac{sqrt{2}}{4} = \frac{\sqrt{6} - \sqrt{2}}{4}.$

By the angle sum formula for tangents, we have:

$\tan 75^\circ = \frac{\tan 45^\circ + \tan 30^\circ}{1 - \tan 45^\circ \tan 30^\circ} = \frac{1+\frac{\sqrt{3}}{3}}{1 - \frac{\sqrt{3}}{3}} = \frac{3 + \sqrt{3}}{3 - \sqrt{3}}$.

Rationalizing the denominator gives $\frac{12+6\sqrt{3}}{6} = 2+\sqrt{3}$ as the final answer.