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

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)