# A triangle has sides A, B, and C. If the angle between sides A and B is (pi)/8, the angle between sides B and C is (2pi)/3, and the length of side B is 5, what is the area of the triangle?

Jul 14, 2017

The area of the triangle is $= 9.52 {u}^{2}$

#### Explanation:

The angles are

$\hat{C} = \frac{1}{8} \pi$

$\hat{A} = \frac{2}{3} \pi$

This is an isoceles triangle.

$\hat{B} = \pi - \left(\frac{1}{8} \pi + \frac{2}{3} \pi\right) = \frac{5}{24} \pi$

$b = 5$

We apply the sine rule to the triangle

$\frac{a}{\sin} \left(\hat{A}\right) = \frac{b}{\sin} \left(\hat{B}\right)$

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

$a = 5 \cdot \sin \frac{\frac{2}{3} \pi}{\sin} \left(\frac{5}{24} \pi\right) = 7.11$

The area of the triangle is

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

$= \frac{1}{2} \cdot 7.11 \cdot 7 \cdot \sin \left(\frac{1}{8} \pi\right)$

$= 9.52$