A triangle has sides A, B, and C. Sides A and B are of lengths 1 and 3, respectively, and the angle between A and B is (pi)/8 . What is the length of side C?

$c = \sqrt{10 - 6 \cos \left(\frac{\pi}{8}\right)} = \sqrt{10 - 3 \cdot \sqrt{2 + \sqrt{2}}}$

$c = 2.1111$ units

Explanation:

By the $\textcolor{b l u e}{\text{Cosine law}}$ the side third side can readily computed when $\textcolor{b l u e}{\text{two sides and an included angle are given}}$

$a = 1$ and $b = 3$ and angle $C = \frac{\pi}{8}$

$c = \sqrt{{a}^{2} + {b}^{2} - 2 \cdot a \cdot b \cdot \cos C}$

$c = \sqrt{{1}^{2} + {3}^{2} - 2 \cdot 1 \cdot 3 \cdot \cos \left(\frac{\pi}{8}\right)}$

$c = \sqrt{10 - 6 \cdot \cos \left(\frac{\pi}{8}\right)}$ and $\cos \left(\frac{\pi}{8}\right) = \frac{1}{2} \sqrt{2 + \sqrt{2}}$

$c = 2.1111$units

Have a nice day !!! from the Philippines..