Angle of Depression: Definition & Formula

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  • 0:05 angle of depression
  • 1:43 Using the Angle
  • 2:23 Formula
  • 3:04 Examples
  • 4:30 Lesson Summary
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Lesson Transcript
Instructor: Beverly Maitland-Frett

Beverly has taught mathematics at the high school level and has a doctorate in teaching and learning.

This lesson will explore angles of depression with reference to angles of elevation. Using real world examples, we will seek to understand the similarities and differences of these angles.

Angle of Depression

Imagine that you are standing about 3 ft. away from a spot on your kitchen floor. As you look down at the spot, imagine a diagonal line is formed from your eyes to the spot. This line is called your line of sight, or the imaginary line that forms between your eye and an object. Now, what if you look out your kitchen window in a straight horizontal line and see someone coming towards you? Would your line of sight change? Yes, it would. However, most of the time we are looking out, not down, therefore that horizontal line is always present. The only thing that does change is whether you're looking down or up. When you were looking at that spot on your kitchen floor, you were looking down. The angle that formed between this diagonal line and the straight horizontal line you used to look out your kitchen window, is called the angle of depression.

Angle of Depression

The angle of depression is the angle between the horizontal and your line of sight (when looking down).

Angle of Depression

Now, pretend that the spot on your kitchen floor could look back up at you. Would it be using the same angle? It certainly would. This angle is called the angle of elevation. This angle is formed when looking up. Since both the angle of depression and angle of elevation are the same, angles 'x' and 'y' in this diagram have the same measure. And since both the top and bottom lines in the diagram are parallel lines, x and y form alternate interior angles, or congruent angles located on different parallel lines cut by a transversal line. Which means x = y.

Therefore, the angle of depression = angle of elevation.

Using the Angle

The angle of depression is a very important angle to certain professionals, not to mention the fact that we form these angles every time we look at something, and without even realizing it. For example, pilots land planes at certain angles of depression. People in watch towers look down at certain angles of depression. A bird swooping down at a certain angle of depression to catch the worm is another common mathematical example.

If you go outside and look up at a bird nest on your roof, you are looking at a certain angle of elevation. If you do not have a good view of the nest, you would probably step back so that you can view at a better angle. This angle depends on the height and the horizontal distance.

Formula

In order to find the angle of depression, we must have some prior understanding of trigonometric ratios, sine, cosine and tangent. Remember, these ratios only apply to right triangles.

Trigonometric Ratios

The main ratio that we use to find the angle of depression is tangent.

The angle of depression may be found by using this formula: tan y = opposite/adjacent. The opposite side in this case is usually the height of the observer or height in terms of location, for example, the height of a plane in the air. The adjacent is usually the horizontal distance between the object and the observer.

Examples

Example 1

Pretend that you are about 5 ft. tall and, as we said earlier, you are standing 3 ft. away from a spot on your kitchen floor. What is the angle of depression? Remember tan y = opposite (your height) divided by adjacent (the distance between you and the spot).

Tangent

Tangent y = 5/3. 'Y' = the inverse of tangent x 5/3.

Tangent Example

As the angle of depression and the angle of elevation are equal, or angles 'x' and 'y,' we need to use a scientific calculator to find the inverse function of the tangent, which gives us the angle of depression:

59.04 degrees.

Example 2

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