Angle of Depression

Part of the Right-Triangle Trigonometry section of Coddy's Trigonometry journey. Lesson 36 of 50.

Look down from a height and the angle between the horizontal and your line of sight is the angle of depression. It sits at the top, outside the triangle formed by the cliff and the ground.

$A 50 m cliff, a boat seen 32° below the horizontal at the top, the angle y° at the boat, and the distance x$

The horizontal at the top is parallel to the sea (the matching chevrons say so), so the angle of depression and the angle of elevation from the boat, y°, are alternate angles: y = 32. Inside the triangle, the 50 m cliff is opposite y° and x is adjacent, so tan(32°) = 50/x and x = 50/tan(32°), about 80.02 m.

On the board that is two facts: Alternate angles for y, then Tangent ratio for x. Never put the angle of depression itself into the triangle at the top: there it sits between the horizontal and the line of sight, and the cliff is not one of its sides.

Challenge

From the top of a 45 m cliff, a boat is seen at an angle of depression of 18°. Find the boat's distance x from the foot of the cliff, to two decimal places. Then press Check.

Try it yourself

$Use the figure to find x$