# A triangle has sides A, B, and C. Sides A and B are of lengths 11 and 12, respectively, and the angle between A and B is (5pi)/8 . What is the length of side C?

Apr 12, 2016

≈ 19.13

#### Explanation:

In a triangle , given 2 sides and the angle between them, to find the 3rd side use the $\textcolor{b l u e}{\text{ cosine rule }}$

$\textcolor{red}{| \overline{\underline{\textcolor{w h i t e}{\frac{a}{a}} \textcolor{b l a c k}{{c}^{2} = {a}^{2} + {b}^{2} - \left(2 a b \cos C\right)} \textcolor{w h i t e}{\frac{a}{a}} |}}}$

where a , b are the 2 known sides and angle C , is the angle between them.
here a = 11 , b = 12 and angle C = $\frac{5 \pi}{8}$

substituting these values into the formula

${c}^{2} = {11}^{2} + {12}^{2} - \left(2 \times 11 \times 12 \times \cos \left(\frac{5 \pi}{8}\right)\right)$

$= 121 + 144 - \left(- 101.03\right) = 366.03$

now  c^2 = 366.03 rArr c = sqrt366.03 ≈ 19.13