# 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 (7pi)/12 . What is the length of side C?

Jun 27, 2017

approximately (to 3 decimal places): $3.400$

#### Explanation:

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

${C}^{2} = {3}^{2} + {1}^{1} - 2 \cdot 3 \cdot 1 \cdot \cos \left(\frac{7 \pi}{12}\right)$

$\textcolor{w h i t e}{\text{XXX")=9+1- 6 * (-0.252914271)color(white)("xxx}}$(calculator use for this and all points beyond)

$\textcolor{w h i t e}{\text{XXX}} = 10 + 1.55291427$

$\textcolor{w h i t e}{\text{XXX}} = 11.55291427$

$C = \sqrt{11.55291427}$

$\textcolor{w h i t e}{\text{XX}} = 3.398957821$