Question

From a balloon 1191 feet high, the angle of depression to the ranger headquarters is 72°54'. How far is the headquarters from a point on the ground directly below the balloon (to the nearest foot)?

Answer 1

If you take a look at my really poorly drawn diagram, we can see it is a trigonometry problem. The angle, however, is outside the triangle, and we can find the angle inside the triangle next to it by taking it away from 90°:
90°0' - 72°54' = 17°6' ≈ 17.1°.
We can find the other angles becuase one is 90° and the other is:
180°-90°-17°6'=72°54' ≈ 72.9°(which you can see if you know your angle theorems).

We now have diagram 2, nicely simplified so we can see where to go.
Using the Sine Rule, i.e.
$\frac{sin(A)}{a} = \frac{sin(B)}{b}$
We can rearrange to find our length b:
$b=a \frac{sinB}{sinA}$
Substituting numbers in:
$b=1191 \frac{sin(17.1)}{sin(72.9)} =366.399...ft$
b=366 feet.