Find the exact value by using a half-angle identity.
-$\tan 75 ^ { \circ }$

Question

Quizplus · Accepted Answer

The exact value of $\tan 75^\circ$ can be found using the half-angle identity for tangent: $\tan \left(\frac{\theta}{2}\right) = \sqrt{\frac{1 - \cos \theta}{1 + \cos \theta}}$. Since $75^\circ = \frac{150^\circ}{2}$, we use $\cos 150^\circ = -\sqrt{3}/2$. Substituting into the half-angle formula gives $\tan 75^\circ = \sqrt{\frac{1 - (-\sqrt{3}/2)}{1 + (-\sqrt{3}/2)}} = \sqrt{\frac{2 + \sqrt{3}}{2 - \sqrt{3}}} = 2 + \sqrt{3}$, after rationalizing the denominator.

Find the Exact Value by Using a Half-Angle Identity tan⁡75∘\tan 75 ^ { \circ }tan75∘

Find the Exact Value by Using a Half-Angle Identity $\tan 75 ^ { \circ }$