# A triangle has sides A, B, and C. Sides A and B are of lengths 6 and 8, respectively, and the angle between A and B is (7pi)/8 . What is the length of side C?

Jun 11, 2018

c=sqrt(100-48sqrt(2+sqrt(2))

#### Explanation:

Note that

$\cos \left(\frac{\pi}{8}\right) = - \frac{1}{2} \cdot \sqrt{2 + \sqrt{2}}$
so we get after the Theorem of cosines

${c}^{2} = {a}^{2} + {b}^{2} - 2 a b \cos \left(\gamma\right)$

$c = \sqrt{36 + 64 - 96 \cdot \cos \left(\frac{\pi}{8}\right)}$
and we get

$c = \sqrt{100 - 48 \sqrt{2 + \sqrt{2}}}$