# A triangle has sides A, B, and C. Sides A and B are of lengths 3 and 1, respectively, and the angle between A and B is pi/6. What is the length of side C?

Feb 26, 2016

Length of $C \approx 2.72$

#### Explanation:

According to the Law of Cosines:
$\textcolor{w h i t e}{\text{XXX}} {C}^{2} = {A}^{2} + {B}^{2} - A B \cos \left(\angle c\right)$
where $\angle c$ is the angle opposite side $C$ i.e. the angle between $A$ and $B$.

So for the given values:
$\textcolor{w h i t e}{\text{XXX}} {C}^{2} = {3}^{2} + {1}^{2} - 3 \cos \left(\frac{\pi}{6}\right)$
$\textcolor{w h i t e}{\text{XXX}} = 10 - \frac{3 \sqrt{3}}{2}$
$\textcolor{w h i t e}{\text{XXX}} \approx 7.401924$

$C \approx \sqrt{7.401924} \approx 2.72$