Question

find the exact value of cos(165)

Answer 1

Find the exact value of $\cos(165)$

Solution:

$$\begin{array}{l} \cos(165) = \cos(180 - 15) = -\cos(15) = \\ = -\cos\left(\frac{30}{2}\right) = -\sqrt{\frac{1 + \cos(30)}{2}} = -\sqrt{\frac{1 + \frac{\sqrt{3}}{2}}{2}} = -\sqrt{\frac{2 + \sqrt{3}}{4}} = -\frac{1}{2}\sqrt{2 + \sqrt{3}} \end{array}$$

Answer:

$$\cos(165) = -\frac{1}{2}\sqrt{2 + \sqrt{3}}$$

This value is approximately $\approx -0.96592583$