# 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?

Jan 4, 2018

$8 \tan \left(\frac{\pi}{12}\right) = 16 - 8 \sqrt{3} \approx 2.146$

#### Explanation:

From the given we know that:

Angle C has measure $\frac{\pi}{2}$ (so we have a right triangle).
Angle A has measure $\frac{\pi}{12}$.

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.

$\tan \left(A\right) = \frac{a}{b}$

$\tan \left(\frac{\pi}{12}\right) = \frac{a}{4} \setminus \rightarrow a = 4 \tan \left(\frac{\pi}{12}\right)$.

Since $a$ and $b$ are the legs of the right triangle the area of the triangle is $\frac{1}{2} a \cdot b$ so the area is $\frac{1}{2} \cdot \left(4 \tan \left(\frac{\pi}{12}\right) \cdot 4\right) = 8 \tan \left(\frac{\pi}{12}\right) = 16 - 8 \sqrt{3} \approx 2.146$.

I used a calculator for the to numerical values.