# A triangle has sides A, B, and C. Sides A and B are of lengths 3 and 1, respectively, and the angle between A and B is (5pi)/12 . What is the length of side C?

Jun 22, 2018

$c = \sqrt{10 - 3 \cdot \frac{\sqrt{3} - 1}{\sqrt{2}}}$

#### Explanation:

By the Theorem of cosines
${c}^{2} = {a}^{2} + {b}^{2} - 2 a b \cos \left(\gamma\right)$
we get

${c}^{2} = 1 + 9 - 6 \cos \left(5 \cdot \frac{\pi}{12}\right)$
Note that $\cos \left(5 \cdot \frac{\pi}{12}\right) = \frac{\sqrt{3} - 1}{2 \sqrt{2}}$
so we get

$c = \sqrt{10 - 3 \cdot \frac{\sqrt{3} - 1}{\sqrt{2}}}$