# A triangle has sides A, B, and C. Sides A and B have lengths of 6 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?

May 2, 2016

Let us plot the triangle, and let us find its area using trigonometry.

#### Explanation:

First of all, our triangle is similar to the next one: Although there are many ways to find the area of a triangle (you can check it on Wikipedia ), we will use trigonometry: The area of the above triangle can be calculated by:

$A = \frac{1}{2} a b \sin \gamma$

i.e. we must multiply two sides and the sine of the angle between them.

We know A and B sides, but we do not know the angle. We may calculate it by knowing that, for any triangle with angles $\alpha , \beta , \gamma$:

$\alpha + \beta + \gamma = \pi \rightarrow$

$\rightarrow \frac{7 \pi}{24} + \frac{13 \pi}{24} + \gamma = \pi \rightarrow$

$\rightarrow \gamma = \frac{4 \pi}{24} = \frac{\pi}{6}$

And now:

$A = \frac{1}{2} \cdot \text{A" cdot "B} \cdot \sin \gamma = \frac{1}{2} \cdot 6 \cdot 4 \cdot \sin \left(\frac{\pi}{6}\right) = \textcolor{b l u e}{6}$