Question

Find the exact value of the expression:

$\cos {165}^{\circ }$

Short answer

$-\frac{1}{4}(\sqrt{6}+\sqrt{2})$

Step-by-step solution

Step 1. Given information 

We have been given an expression $\cos {165}^{\circ }$.

We have to find its exact value.

Step 2. Simplify the expression

We can also write the expression as:

$\cos {165}^{\circ }=\cos ({135}^{\circ }+{30}^{°})$

Step 3.  Apply the compound angle formula

$\cos ({135}^{\circ }+{30}^{°})$has the form of $\cos (\alpha +\beta )$

Therefore, apply

$\cos \left(\alpha +\beta \right)=\cos \alpha \cos \beta -\sin \alpha \sin \beta$

$\begin{array}{r}\cos {135}^{\circ }\cos {30}^{\circ }-\sin {135}^{\circ }\sin {30}^{\circ }=\left(-\left(\frac{\sqrt{2}}{2}\right)\cdot \left(\frac{\sqrt{3}}{2}\right)\right)-\left(\left(\frac{\sqrt{2}}{2}\right)\cdot \left(\frac{1}{2}\right)\right)\\ =\left(-\frac{\sqrt{6}}{4}\right)-\left(\frac{\sqrt{2}}{4}\right)\\ =\left(\frac{-\sqrt{6}-\sqrt{2}}{4}\right)\\ =-\frac{1}{4}(\sqrt{6}+\sqrt{2})\end{array}$