# A triangle has sides A, B, and C. Sides A and B have lengths of 3 and 5, respectively. The angle between A and C is (3pi)/8 and the angle between B and C is  (5pi)/12. What is the area of the triangle?

Jul 23, 2017

The area of the triangle is $= 4.57 {u}^{2}$

#### Explanation:

The lengths of the sides are

$a = 3$

$b = 5$

The angles are

$\hat{A} = \frac{5}{12} \pi$

$\hat{B} = \frac{3}{8} \pi$

Therefore,

$\hat{C} = \pi - \left(\frac{5}{12} \pi + \frac{3}{8} \pi\right) = \pi - \frac{19}{24} \pi = \frac{5}{24} \pi$

The area of the triangle is

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

$= \frac{1}{2} \cdot 3 \cdot 5 \cdot \sin \left(\frac{5}{24} \pi\right)$

$= 4.57 {u}^{2}$