# A triangle has sides A, B, and C. Sides A and B are of lengths 9 and 8, respectively, and the angle between A and B is (11pi)/12 . What is the length of side C?

Apr 8, 2018

color(indigo)(c = 16.85 " units"

#### Explanation:

As per the Law of Cosines,

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

${c}^{2} = {9}^{2} + {8}^{2} - 2 \cdot 9 \cdot 8 \cdot \cos \left(\frac{11 \pi}{12}\right)$

${c}^{2} = 81 + 64 + 144 \cos \left(\frac{\pi}{12}\right) , \text{ as } \cos \left(\pi - \theta\right) = - \cos \theta$

${c}^{2} = 145 + 144 \cos \left(\frac{\pi}{12}\right) = 284.09$

$c = \pm \sqrt{284.09} = 16.85$

As c cannot be negative, color(indigo)(c = 16.85 " units"