# 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 37, what is the area of the triangle?

Jan 26, 2018

About $183.41 {\text{units}}^{2}$ (exact answer below).

#### Explanation:

A given angle is $\frac{\pi}{2}$. To convert from radians to degrees, you must multiply by the conversion factor (180º)/pi:

cancel(pi)/2*(180º)/cancel(pi) = 90º

Now we now that this triangle is a right triangle because it contains a 90º angle. Another given angle is $\frac{\pi}{12}$, so we do the same conversion:

cancel(pi)/12*(180º)/cancel(pi)=(180º)/12 = 15º

The tangent trigonometric function $\tan \left(\theta\right)$ is defined as the ratio between the side opposite the angle $\theta$ and the side adjacent to $\theta$. So we can say that:

tan(15º) = A/B

And since $B = 37$:

tan(15º) = A/37

Now we can solve for A, the height of the triangle, and get:

A=37*tan(15º)~~9.91

Now we can find the area of our triangle by using the formula for the area of triangles and plugging in our values:

${A}_{\text{triangle}} = \frac{b \cdot h}{2}$

= (37*(37*tan(15º)))/2

=(37^2*tan(15º))/2

=(1369*tan(15º))/2

$\approx 183.41$