The angle you lower your eyes

You stand on a cliff and spot a boat below. How far out is it?

The angle of depression

Distance = cliff height ÷ tan of the angle of depression.

The angle of depression is the angle you look down.

At the cliff top, the horizontal runs level out to sea. You look down to the boat. The angle of depression opens down from the horizontal.

The same angle at the boat

The horizontal at the top and the water are parallel lines. The sight line cuts across both. Alternate angles lie between the parallel lines. They are on opposite sides of the cutting line. Because the lines are parallel, alternate angles are equal. So the depression at the top equals the elevation at the boat.

Rule: On level ground, depression from the top = elevation from the object.

The boat, the cliff foot and the top make a right triangle. The angle at the boat equals the angle of depression. The cliff faces that angle, so the cliff is opposite. The distance is adjacent. Tan of the angle = height ÷ distance, so distance = height ÷ tan.

Sin of the angle = height ÷ line of sight. The line of sight is the longest side. Line of sight = height ÷ sin of the angle.

At 30°: 30² + (30√3)² = 900 + 2700 = 3600. And 60² = 3600. The Pythagoras check holds.

Try the lab to finish this lesson.