Question

A triangle has sides A, B, and C. Sides A and B have lengths of 7 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?

Answer 1

`"Area"_triangle=~~12.97`

Explanation:

Call the angle opposite side A as `/_a` (i.e. the angle between B and C);
and similarly the angle opposite Side B as `/_b`
and the angle opposite Side C as `/_c`

We are told (among other things) that:
`A=7, /_a=(7pi)/24, and /_c=(11pi)/24`

The Sine Law tells us
`color(white)("XXX")C/sin(c) = A/sin(a)`

So
`color(white)("XXX")C=7/(sin((7pi)/24))*sin((11pi)/24)`

Evaluating we get: `C=11`

We now have the lengths of the three sides and can apply Heron's Formula:
`color(white)("XXX")"Area"_triangle = sqrt(S(A-A)(S-B)(S-C))`
where `S` is the semi-perimeter (i.e. `(A+B+C)/2`)