# A triangle has sides A, B, and C. The angle between sides A and B is (7pi)/12 and the angle between sides B and C is pi/12. If side B has a length of 26, what is the area of the triangle?

Aug 5, 2017

$\text{area "~~ 97.568" square units}$

#### Explanation:

$\text{calculate the area ( A ) of the triangle using}$

•color(white)(x)A=1/2ABsinC

$\text{where C is the angle between sides A and B}$

$\text{we require to calculate the side A}$

$\text{ the third angle in the triangle is}$

$\pi - \left(\frac{7 \pi}{12} + \frac{\pi}{12}\right) = \frac{\pi}{3}$

$\text{using the "color(blue)"sine rule }$in triangle ABC

$\frac{A}{\sin} \left(\frac{\pi}{12}\right) = \frac{26}{\sin} \left(\frac{\pi}{3}\right)$

$\Rightarrow A = \frac{26 \sin \left(\frac{\pi}{12}\right)}{\sin \left(\frac{\pi}{3}\right)} \approx 7.77$

$\Rightarrow \text{area( A)} = \frac{1}{2} \times 7.77 \times 26 \times \sin \left(\frac{7 \pi}{12}\right)$

$\textcolor{w h i t e}{\Rightarrow a r e a A} \approx 97.568 \text{ to 3 dec. places}$