# A triangle has sides A, B, and C. Sides A and B have lengths of 5 and 7, respectively. The angle between A and C is (pi)/12 and the angle between B and C is  (2pi)/3. What is the area of the triangle?

Feb 10, 2016

Area is $12.3745$.

#### Explanation:

Formula for area of a triangle in terms of its two sides is half the product of two sides multiplied by sine of the angle between the two sides.

Here we have sides A and B, but we do not know the angle between them.

Fr this, as sum of internal angles of a triangle is $\pi$ and other two angles are $\frac{\pi}{12}$ and "2pi/3#,

third angle is $\pi - \frac{\pi}{12} - \frac{2 \pi}{3}$ i.e. $\frac{3 \pi}{12}$ or $\frac{\pi}{4}$

Hence, as $\sin \left(\frac{\pi}{2}\right)$ is $\left(\frac{1}{\sqrt{2}}\right)$

Area of triangle is $\left(\frac{1}{2}\right) \cdot 5 \cdot 7 \cdot \sin \left(\frac{\pi}{4}\right)$

or $\left(\frac{1}{2}\right) \cdot 5 \cdot 7 \cdot \left(\frac{1}{\sqrt{2}}\right)$,

i.e. $12.3745$