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 37, what is the area of the triangle?

Answer 1

About `183.41 "units"^2` (exact answer below).

Explanation:

A given angle is `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 `pi/12`, so we do the same conversion:

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

The tangent trigonometric function `tan(theta)` 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_"triangle" = (b*h)/2`

`= (37*(37*tan(15º)))/2`

`=(37^2*tan(15º))/2`

`=(1369*tan(15º))/2`

`~~183.41`