# A triangle has sides A, B, and C. Sides A and B are of lengths 4 and 12, respectively, and the angle between A and B is pi/3. What is the length of side C?

Feb 5, 2018

See explanation.

#### Explanation:

If you have 2 sides of a triangle given and the angle between them, you can use the Cosine Theorem to calculate the remaining side:

## ${C}^{2} = {A}^{2} + {B}^{2} - 2 A B \cos c$

where $c$ is the angle opposite to side $C$ (i.e. angle between $A$ and $B$)

If we apply the given data we have:

${C}^{2} = {4}^{2} + {12}^{2} - 2 \cdot 4 \cdot 12 \cdot \cos \left(\frac{\pi}{3}\right)$

${C}^{2} = 16 + 144 - 96 \cdot \frac{1}{2}$

${C}^{2} = 160 - 48 = 112$

$C = \sqrt{112} = 4 \sqrt{7}$