From a balloon 734 feet high, the angle of depression to the ranger headquarters is 70°31'. How far is the headquarters from a point on the ground directly below the balloon(to the nearest foot)?

You can draw a right triangle with the first point (A) as the balloon, the second point (B) the HQ, and the third point (C) directly above HQ, to the side of the balloon.

The angle BAC is 70°31'. This is (70+31/60)° ≈ 70.5167°

Side BC is 734.

The hypotenuse AB is unknown. We will call it x.

Side BC is opposite the angle. We know that:

sinθ=opposite/hypotenuse

So we can say:

sin(70.5167)=734/x

x*sin(70.5167)=734 (multiply both sides by x)

x=734/(sin(70.5167)) (divide both sides by sin(70.5167)

Put in into the calculator for 778.5826≈ 779 ft