Angle of Elevation: Definition, Formula & Examples

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Lesson Transcript
Instructor: Eric Istre

Eric has taught high school mathematics for more than 20 years and has a master's degree in educational administration.

This lesson will define an angle of elevation, and it will provide some basic skills necessary to calculate the measure of one of these types of angles or to use it to calculate another value. Two example problems are also provided.

What Is an Angle of Elevation?

Have you ever looked up to see a plane passing overhead? Have you ever stood in a field at night and glanced up at the moon? Have you ever walked down a city sidewalk and turned your view upward toward the tops of the buildings? I'm sure you've done at least one of these. This means that you have experienced an angle of elevation firsthand.

An angle of elevation is simply the amount that you would have turned your view upward from the horizontal to see the plane, the moon, or the tops of the buildings. You may be wondering what is meant by the horizontal. The horizontal in these situations would be looking straight ahead. In other words, if you alter your line of sight from being straight ahead to looking upward, then you have created an angle of elevation. It is important to note that an angle of elevation is similar to an angle of depression, and these two angles are congruent (have the same measurement) when describing the view between the same two objects.

Now, there are a few other tidbits that will be important to know before working on a problem. You need to know the three basic trigonometric (trig) ratios:

sine = opposite/hypotenuse

These three ratios are abbreviated as sin, cos, and tan.

Also, when doing calculations for angle of elevation, your answer will have degrees for the unit. If you are using an advanced calculator, you will need to make sure that it is in degree mode. Failing to do this will result in incorrect answers.

Calculating the Angle of Elevation

Let's use one of the scenarios from before. You are looking up into the sky and notice a plane. This is represented by the diagram seen below.

Notice that the details will create a right triangle. This is what allows us to use the basic trig relationships. Suppose that you know the height of the plane above the ground (500 feet) and the distance along the ground from the observer to the point on the ground directly below the plane (1200 feet). What is the angle of elevation?

First, the angle itself is your reference point. With respect to the angle of elevation, the opposite side and the adjacent side are given. Which one of the trig relationships has the opposite and adjacent? That's right! We need to use the tangent.

Second, use the following information to create an equation. We will use angle A to represent the angle of elevation.

tan (angle A) = opposite / adjacent

tan (angle A) = 500 / 1200

tan (angle A) = 0.4167

Normally, we round this value to the nearest ten-thousandth (4th decimal place).

If you are looking for the measure of the angle, you must use the inverse function on the calculator (tan^-1). This is also referred to as the arctangent (arctan).

angle A = tan^-1 (0.4167) = 22.6 degrees

Degree measurements are usually rounded to the nearest tenth (1st decimal place).

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