# A triangle has sides A, B, and C. Sides A and B are of lengths 6 and 1, respectively, and the angle between A and B is (7pi)/12 . What is the length of side C?

Jul 25, 2016

C=sqrt(37+3(sqrt(6)-sqrt(2))

#### Explanation:

You can apply the theorem of Carnot, by which you can calculate the lenght of the third side C of a triangle if you know two sides, A and B, and the angle $\hat{A B}$ between them:

${C}^{2} = {A}^{2} + {B}^{2} - 2 \cdot A \cdot B \cdot \cos \left(\hat{A B}\right)$

Then ${C}^{2} = {6}^{2} + {1}^{2} - 2 \cdot 6 \cdot 1 \cdot \cos \left(\frac{7 \pi}{12}\right)$

${C}^{2} = 36 + 1 - 12 \cdot \left(- \frac{1}{4} \left(\sqrt{6} - \sqrt{2}\right)\right)$

$= 37 + 3 \left(\sqrt{6} - \sqrt{2}\right)$

C=sqrt(37+3(sqrt(6)-sqrt(2))