# A triangle has sides A, B, and C. Sides A and B are of lengths 8 and 4, respectively, and the angle between A and B is (11pi)/12 . What is the length of side C?

Dec 18, 2015

$C \cong 11.9$

#### Explanation:

If we denote the angle between $A$ and $B$ as $\angle C$ (not to be confused with the side $C$ opposite $\angle C$)

The Law of Cosines tells us
$\textcolor{w h i t e}{\text{XXX}} {C}^{2} = {A}^{2} + {B}^{2} - 2 A B \cdot \cos \left(\angle C\right)$
$\Rightarrow$
$\textcolor{w h i t e}{\text{XXX}} C = \sqrt{{A}^{2} + {B}^{2} - 2 A B \cdot \cos \left(\angle C\right)}$

We are told that
$\textcolor{w h i t e}{\text{XXX}} A = 8$
$\textcolor{w h i t e}{\text{XXX}} B = 4$
$\textcolor{w h i t e}{\text{XXX}} \angle C = \frac{11 \pi}{12}$

Plugging these values into the formula gives
$\textcolor{w h i t e}{\text{XXX}} C = 11.90878889$