# A triangle has sides A, B, and C. Sides A and B have lengths of 4 and 1, respectively. The angle between A and C is (3pi)/8 and the angle between B and C is  (5pi)/12. What is the area of the triangle?

Oct 5, 2017

3.2245

#### Explanation:

The clue here is to realize that you can split up this triangle into two right triangles. Then, by using the appropriate trig identities, you can find the lengths needed to compute the area of both triangles. Adding both areas will then give you the total area.

Let's first work with the "A" side of the triangle:

$\cos \left(\frac{3 \pi}{8}\right) = \text{base"/4, "base} = 1.5307$
$\sin \left(\frac{3 \pi}{8}\right) = \text{height"/4, "height} = 3.6955$

The area of one side of this triangle, then, is:

$A = \frac{1}{2} \cdot b \cdot h = \frac{1}{2} \cdot 1.5307 \cdot 3.6955 = 2.8284$

Now let's compute the area of the other side. Since we know the height, which is universal for both triangles, we just need to compute the base.

$\cos \left(\frac{5 \pi}{12}\right) = \text{base"/1, "base} = 0.2588$

The area of the other side of this triangle, then, is:

$A = \frac{1}{2} \cdot b \cdot h = \frac{1}{2} \cdot 1.5307 \cdot 0.2588 = 0.3961$

Thus, the total area is:

$A = 0.2588 + 0.3961 = 3.2254$