# A triangle has sides A, B, and C. Sides A and B have lengths of 2 and 4, respectively. The angle between A and C is (7pi)/24 and the angle between B and C is  (13pi)/24. What is the area of the triangle?

Jan 10, 2018

2

#### Explanation:

We know:
$\rightarrow$ angle $A = \frac{13 \pi}{24}$
$\rightarrow$ angle $B = \frac{7 \pi}{24}$

therefore we know angle $C = \frac{4 \pi}{24} = \frac{\pi}{6}$ because the angles must sum to $\pi$.

We also know:
$\rightarrow$ side $a = 2$
$\rightarrow$ side $b = 4$

So we know angle $C$ and the sides adjacent to it, $a$ and $b$. We can find the area using:

$A r e a = \frac{1}{2} a \cdot b \setminus \cdot \sin \left(C\right)$

$= \frac{1}{2} \left(2\right) \left(4\right) \sin \left(\frac{\pi}{6}\right) = 4 \sin \left(\frac{\pi}{6}\right)$

since $\sin \left(\frac{\pi}{6}\right) = \frac{1}{2}$, we get:

Area $= \frac{1}{2} \left(4\right) = 2$