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 4, what is the area of the triangle?

Answer 1

`8tan(pi/12) = 16-8sqrt(3) approx 2.146`

Explanation:

From the given we know that:

Angle C has measure `pi/2` (so we have a right triangle).
Angle A has measure `pi/12`.

We know that side b has length 4.

It's helpful to draw a right triangle with hypotenuse c, right angle C, and the rest of the given information filled in.

Since side b is adjacent to angle A we can use tangent to find side a.

`tan(A) = a/b`

`tan(pi/12) = a/4\rightarrow a = 4tan(pi/12)`.

Since `a` and `b` are the legs of the right triangle the area of the triangle is `1/2a*b` so the area is `1/2*(4tan(pi/12)*4) =8tan(pi/12) = 16-8sqrt(3) approx 2.146`.

I used a calculator for the to numerical values.