medium difficulty7 min

Angle of depression

The angle of depression is the angle made by the line of sight with the horizontal when an object is seen below eye level.

In one line: Depression means going down, so the line of sight goes below the horizontal.

Practice This Concept

Understand it

When you stand at a window, on a terrace, or on a bridge and look down at an object, your line of sight goes downward from the horizontal. The downward angle between the horizontal line at your eye and the slant line to the object is called the angle of depression. In diagrams, it is often equal to the angle of elevation from the lower point because of alternate interior angle reasoning on parallel horizontals.

Example

A person standing on a cliff looks down at a boat in the sea. The angle between the horizontal through the person's eye and the line of sight to the boat is the angle of depression.

Formula and rule

Angle of depression = angle between the horizontal through the observer and the downward line of sight.

See it worked out

Follow the example line by line, then use the same method on new questions.

Worked example

1From the top of a lighthouse, a guard looks down at a boat.
2The angle between the horizontal and the line of sight downward is the angle of depression, and it can be matched with the boat's angle of elevation in a parallel-line diagram.

Method you can reuse

1Find the observer at the higher point.
2Draw the horizontal through the eye.
3Draw the downward line of sight.
4Measure the angle between the horizontal and the downward line.
5Name it as angle of depression.

Sketch it in your answer

Draw the observer on top, a horizontal line through the eye, and a slant line going downward to the object. Mark the angle at the top.

Check yourself

Answer the question in your head before opening the answer.

Try it yourself

A person on a terrace looks down at a car on the road. Which angle is formed at the observer's eye?

Think first, then show answer

It is the angle of depression, because the object is below the observer's eye level and the line of sight is below the horizontal.

Common wrong answer

A common wrong answer is to measure from the vertical drop. That gives a different angle and breaks the standard trigonometry setup.

Don't mix this up

Students often measure the angle from the vertical drop line or from the ground, which gives the wrong angle.

Exam tip

First draw the horizontal at the observer's eye level. For depression, the line of sight goes downward below that line.