# A triangle has sides A, B, and C. Sides A and B are of lengths 7 and 6, respectively, and the angle between A and B is (5pi)/8 . What is the length of side C?

Jan 1, 2016

This problem may be solved easily by using the law of cosines.

#### Explanation:

Given a triangle, with sides $a$, $b$ and $c$, and angles $\alpha$, $\beta$ and $\gamma$ (best see the figure below), we can calculate the length of a given side by knowing the lengths of the other two, and the angle between them.

For example, to calculate the length of $c$:

${c}^{2} = {a}^{2} + {b}^{2} - 2 a b \cos \gamma$

In our case:

${c}^{2} = {7}^{2} + {6}^{2} - 2 \cdot 7 \cdot 6 \cdot \cos \frac{5 \pi}{8} = 117.15$
$\rightarrow c = \sqrt{117.15} = 10.82$