Chapter 7.5, Problem 15AYU

To determine

To find: The exactvalue of the given expression.

Answer to Problem 15AYU

The exact value of $\cos {165}^{\circ }$ is $-\frac{\left(1+\sqrt{3}\right)}{\left(2\sqrt{2}\right)}$.

Explanation of Solution

Given:

  $\cos {165}^{\circ }$

The exact value ofthe expression $\cos {165}^{\circ }$ is

  $\begin{array}{l}\cos {165}^{\circ }\\ \text{=}\cos {\left(120+45\right)}^{\circ }\\ \text{=}\cos 120.\cos 45-\sin 120.\sin 45\\ =\left(-\frac{1}{2}\right).\frac{\sqrt{2}}{2}-\frac{\sqrt{3}}{2}.\frac{\sqrt{2}}{2}\end{array}$

  $\begin{array}{l}=\frac{\sqrt{2}}{2}\left(\frac{1}{2}-\frac{\left(\sqrt{3}\right)}{\left(2\right)}\right)\\ =-\frac{\left(1+\sqrt{3}\right)}{\left(2\sqrt{2}\right)}\end{array}$

Hence, the exact value of $\cos {165}^{\circ }$ is $-\frac{\left(1+\sqrt{3}\right)}{\left(2\sqrt{2}\right)}$.