Question

use the half - angle formulas to find the exact value of the expression \\( \cos 165 ^ { \circ } \\).
\\( \cos 165 ^ { \circ } = \square \\)
(type an exact answer, using radicals as needed. use integers or fractions for any numbers in the ex

Explanation:

Step1: Identify the half - angle formula

The half - angle formula for cosine is \(\cos\frac{\alpha}{2}=\pm\sqrt{\frac{1 + \cos\alpha}{2}}\). Since \(165^{\circ}=\frac{330^{\circ}}{2}\) and \(165^{\circ}\) is in the second quadrant (\(\cos\theta<0\) in the second quadrant), we use \(\cos\frac{\alpha}{2}=-\sqrt{\frac{1+\cos\alpha}{2}}\) with \(\alpha = 330^{\circ}\).

Step2: Find the value of \(\cos\alpha\)

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

Step3: Substitute into the half - angle formula

Substitute \(\cos\alpha=\frac{\sqrt{3}}{2}\) into \(\cos\frac{\alpha}{2}=-\sqrt{\frac{1+\cos\alpha}{2}}\).

$$
LATEXBLOCK0
$$

Answer:

\(-\frac{\sqrt{2+\sqrt{3}}}{2}\)