# A triangle has sides A, B, and C. Sides A and B are of lengths 14 and 12, respectively, and the angle between A and B is (5pi)/8 . What is the length of side C?

Mar 10, 2016

≈ 21.647 units

#### Explanation:

Given a triangle with 2 sides and the angle between them known, find the 3rd side using the $\textcolor{b l u e}{\text{ cosine rule }}$

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

where c is the side to be found , a and b are the known sides and $\theta \text{ is the angle between them }$

here a = 14 , b = 12 and $\theta = \frac{5 \pi}{8}$

hence ${c}^{2} = {14}^{2} + {12}^{2} - \left(2 \times 14 \times 12 \cos \left(\frac{5 \pi}{8}\right)\right)$

$= 196 + 144 + 128.582 \ldots$ ≈ 468.582

rArr c = sqrt468.582 ≈ 21.647 " units "