# 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 (pi)/3 and the angle between B and C is  (pi)/4. What is the area of the triangle?

Jan 19, 2016

$A \approx 6.54$

#### Explanation:

The area of a triangle is given by $\frac{1}{2} \cdot B a s e \cdot h e i g h t$
The height $h$ in this example can be found from
$\sin \left(\frac{\pi}{3}\right) = \frac{h}{A} = \frac{h}{3}$
$h = 3 \sin \left(\frac{\pi}{3}\right)$

The base $C = x + y$ and $x$ and $y$ can also be found from trigonometric functions.
$\frac{x}{A} = \cos \left(\frac{\pi}{3}\right)$ so $x = 3 \cos \left(\frac{\pi}{3}\right)$
$\frac{y}{B} = \cos \left(\frac{\pi}{4}\right)$ so $y = 5 \cos \left(\frac{\pi}{4}\right)$

Area $A = \frac{1}{2} \cdot \left(3 \cos \left(\frac{\pi}{3}\right) + 5 \cos \left(\frac{\pi}{4}\right)\right) \cdot 3 \sin \left(\frac{\pi}{3}\right)$

$A = \frac{1}{2} \cdot \left(3 \cdot 0.5 + 5 \cdot 0.7071\right) \cdot 3 \cdot 0.866$
$A \approx 6.54$