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

Apr 7, 2016

≈ 11.745 units

#### Explanation:

In this situation where we have a triangle with 2 sides and the angle between them known , and we wish to find the 3rd side , then we use the $\textcolor{b l u e}{\text{ cosine rule }}$

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

where a and b are the 2 given sides and C , the angle between them. c , is the side to be found.

here a = 12 , b = 5 and C $= \frac{5 \pi}{12}$

substitute these values into the $\textcolor{b l u e}{\text{ cosine rule }}$

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

$= 144 + 25 - \left(120 \cos \left(\frac{5 \pi}{12}\right)\right) = 169 - \left(31.058\right)$

now  c^2 = 137.942 rArr c = sqrt137.942 ≈ 11.745 " units "