Question

A triangle has sides A, B, and C. The angle between sides A and B is `(pi)/2` and the angle between sides B and C is `pi/12`. If side B has a length of 7, what is the area of the triangle?

Answer 1

`S = 49(1-sqrt(3)/2)~=6.5648`

Explanation:

It is a right triangle with catheti `A` and `B` and hypotenuse `C`.
Knowing the length of `B=7` and an angle between `B` and `C` equaled `pi/12`, we can calculate `A` as follows.

Since `A/B = tan(pi/12)`,
it follows that `A = B*tan(pi/12)`

Area of a triangle is
`S =1/2(A*B) = 1/2B^2tan(pi/12)`

The latter can be simplified using a formula
`tan(phi) = (1-cos(2phi)) / sin(2phi)`
from which follows:
`tan(pi/12) = (1-cos(pi/6)) / sin(pi/6) = [1-sqrt(3)/2]/(1/2) = 2-sqrt(3)`

Therefore, the area of a triangle is
`S = (1/2)7^2(2-sqrt(3))=49(1-sqrt(3)/2)`