Question

use a half - angle formula to find the exact value of \\( \tan 165 ^ { \circ } \\).
\\( \tan 165 ^ { \circ } = \\)

Explanation:

Step1: Identify the half - angle formula

The half - angle formula for tangent is \(\tan\frac{\alpha}{2}=\frac{1 - \cos\alpha}{\sin\alpha}\).
Since \(165^{\circ}=\frac{330^{\circ}}{2}\), here \(\alpha = 330^{\circ}\).

Step2: Find the values of \(\sin330^{\circ}\) and \(\cos330^{\circ}\)

We know that \(\sin330^{\circ}=\sin(360^{\circ}- 30^{\circ})=-\sin30^{\circ}=-\frac{1}{2}\) and \(\cos330^{\circ}=\cos(360^{\circ}-30^{\circ})=\cos30^{\circ}=\frac{\sqrt{3}}{2}\).

Step3: Substitute into the half - angle formula

Substitute \(\sin330^{\circ}=-\frac{1}{2}\) and \(\cos330^{\circ}=\frac{\sqrt{3}}{2}\) into \(\tan\frac{\alpha}{2}=\frac{1 - \cos\alpha}{\sin\alpha}\).
\(\tan165^{\circ}=\frac{1-\frac{\sqrt{3}}{2}}{-\frac{1}{2}}=\frac{\frac{2 - \sqrt{3}}{2}}{-\frac{1}{2}}=-(2-\sqrt{3})=\sqrt{3}-2\)

Answer:

\(\tan165^{\circ}=\sqrt{3}-2\)