The angle of depression from the top of the lighthouse down to Joey is 38 degrees. What is the direct distance from the TOP of the lighthouse to Joey's boat if the lighthouse is 40 feet tall? *

Answer 1

top to the boat is the hypotenuse of a right triangle where cos(38) = 40/hypotenuse

distance to the boat from the top = 40/cos38

Answer 2

Consider the triangle made up of the top of the lighthouse, the bottom of the lighthouse, and the boat.

Note that, if the angle of depression is 38°, the angle between the hypotenuse and the lighthouse is 90º - 38° = 52°

40 ft., the height of the lighthouse is the side adjacent to the 52° angle, and the side from the top of the lighthouse to the boat is the hypotenuse.

Then, to get the distance from the top of the lighthouse to the boat, as

height of the lighthouse * distance from the top of the lighthouse to the boat/ height of the lighthouse equals the distance from the top of the lighthouse to the boat, note that

distance from the top of the lighthouse to the boat/ height of the lighthouse equals hypotenuse/adjacent =

sec 52°.

Therefore, distance from the top of the lighthouse to the boat equals 40 * sec 52° = 64.97 feet.