Question

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?

Answer 1

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:

![https://en.wikipedia.org/wiki/Triangle#/media/File:Triangle.TrigArea.svg]()

The area of the above triangle can be calculated by:

`A = 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 {7 pi}/24 + {13 pi}/24 + gamma = pi rightarrow`

`rightarrow gamma = {4 pi}/24 = pi/6`

And now:

`A = 1/2 cdot "A" cdot "B" cdot sin gamma = 1/2 cdot 6 cdot 4 cdot sin (pi/6) = color(blue) 6`