Question

use a half - angle identity to find the exact value.
cos 105°

cos 105° =
(simplify your answer, including any radicals. use integers or fractions for any numbers in the expression.)

Explanation:

Step1: Determine the half - angle formula

The half - angle formula for cosine is $\cos\frac{\alpha}{2}=\pm\sqrt{\frac{1 + \cos\alpha}{2}}$. Since $105^{\circ}=\frac{210^{\circ}}{2}$ and $105^{\circ}$ is in the second quadrant where cosine is negative.

Step2: Substitute $\alpha = 210^{\circ}$ into the formula

We know that $\cos210^{\circ}=-\frac{\sqrt{3}}{2}$. Substituting into the formula $\cos105^{\circ}=-\sqrt{\frac{1+\cos210^{\circ}}{2}}$.

Step3: Simplify the expression inside the square root

$$
LATEXBLOCK0
$$

Step4: Find the value of $\cos105^{\circ}$

Since $\cos105^{\circ}=-\sqrt{\frac{2 - \sqrt{3}}{4}}=-\frac{\sqrt{2-\sqrt{3}}}{2}$

Answer:

$-\frac{\sqrt{2-\sqrt{3}}}{2}$