An observer in a lighthouse 350 feet above sea level observes two ships directly to the west. The angles of depression to the ships are 5º and 8.2º. How far apart are the ships (to the nearest foot)?

Question

xkehaulanix · Answer

The angle of depression is the angle below the horizontal, so using that you can find the angle included in the right triangle. This one's a bit tricky without a picture, so I recommend you try making one. 
The first ship is 5º below the horizontal with respect to the lighthouse. Therefore the angle from the observer to the ship is 85º.
If you set tan(85º)= x/350, you can solve for x.
x= 350 x tan(85º) = 4000.518 ft from the base of the lighthouse

The second ship is 8.2º below the horizontal, so the angle from the observer to the ship is 81.8º. You can use the same setup from before to solve for the second distance.
tan(81.8º) = y/350
y= tan(81.8º) x 350 = 2428.832

Find the difference: 4000.518 - 2428.832 = 1572 feet apart

You could also solve this using two flipped right triangles based on the original angle values...but this way is just easier to picture for me x]