# A triangle has sides A, B, and C. Sides A and B have lengths of 2 and 5, respectively. The angle between A and C is (11pi)/24 and the angle between B and C is  (7pi)/24. What is the area of the triangle?

Mar 15, 2018

The triangle does not exist...
Here's the area anyways:
$A = \frac{5 \sqrt{2}}{2} u {n}^{2} \approx 3.54 u {n}^{2}$

#### Explanation:

Area of the triangle:
$A = \frac{1}{2} a \cdot b \cdot \sin C$
Let's determine angle C:
$\frac{24 \pi}{24} - \frac{7 \pi}{24} - \frac{11 \pi}{24} = \frac{\pi}{4}$

Area of the triangle:
$A = \frac{1}{2} \left(2\right) \left(5\right) \sin \left(\frac{\pi}{4}\right) =$
$A = \frac{5 \sqrt{2}}{2} u {n}^{2} \approx 12.37 u {n}^{2}$

How do I know this triangle particularly does not exist?
Well:
The law of sines states:
SinA/a= SinB/b
Therefore:
$B = \arcsin \left(\frac{S \in A \cdot b}{a}\right)$
$B = \arcsin \left(\frac{S \in \left(\frac{7 \pi}{24}\right) \cdot 5}{2}\right) = u n \mathrm{de} f .$
Since Angle B cannot be computed, this triangle does not exist