# A triangle has sides A, B, and C. Sides A and B are of lengths 1 and 6, respectively, and the angle between A and B is (5pi)/6 . What is the length of side C?

Jun 25, 2018

$c = \sqrt{37 + 6 \sqrt{3}}$

#### Explanation:

We use the Theorem of cosines:

${c}^{2} = {a}^{2} + {b}^{2} - 2 a b \cos \left(\gamma\right)$
and that
$\cos \left(\frac{5}{6} \cdot \pi\right) = - \frac{\sqrt{3}}{2}$
since
$b = 1 , a = 6$

we get

${c}^{2} = 1 + 36 - 2 \cdot 6 \cdot \left(- \frac{\sqrt{3}}{2}\right)$
this is
$c = \sqrt{37 + 6 \cdot \sqrt{3}}$