# A triangle has sides A, B, and C. Sides A and B are of lengths 1 and 2, respectively, and the angle between A and B is (2pi)/3 . What is the length of side C?

Mar 17, 2018

color(brown)(c = sqrt 7

#### Explanation:

$a = 1 , b = 2 , \hat{C} = \frac{2 \pi}{3}$

Applying Law of Cosine,

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

${c}^{2} = {1}^{2} + {2}^{2} - \left(2 \cdot 1 \cdot 2 \cdot \cos \left(\frac{2 \pi}{3}\right)\right)$

${c}^{2} = 5 - 4 \cos \left(\frac{2 \pi}{3}\right) = 5 + 4 \cos \left(\frac{\pi}{3}\right)$ as $\cos \left(\frac{2 \pi}{3}\right) = - \cos \left(\frac{\pi}{3}\right)$ and cos is negative in II quadrant.

${c}^{2} = 5 + \left(4 \cdot \left(\frac{1}{2}\right)\right) = 5 + 2 = 7$

color(brown)(c = sqrt 7