Problem-Solving with Angles of Elevation & Depression

Problem-Solving with Angles of Elevation & Depression
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  • 0:00 Elevation and Depression
  • 1:16 Example Problem 1
  • 3:14 Example Problem 2
  • 4:20 Lesson Summary
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Lesson Transcript
Instructor: Elizabeth Foster

Elizabeth has been involved with tutoring since high school and has a B.A. in Classics.

In this lesson, you'll explore the difference between angles of elevation and depression. You'll also learn how to solve problems related to angles of elevation and depression using basic trigonometric relationships

Elevation and Depression

Quick! What do crashing airplanes, ski lifts, and looking at people from the top of a hill all have in common?

The answer: you can make mathematical models of all of them with angles of elevation and depression! If a person is standing on the ground, looking up at something in the sky, the angle of elevation is the angle formed between the person's line of sight and the ground. For a person looking down at something from a higher point, the angle of depression is the angle between the person's line of sight and an imaginary horizontal line parallel to the ground.

In this lesson, you'll solve some real-world problems with angles of elevation and depression.

We'll also be using the basic trigonometric relationships, abbreviated SOH-CAH-TOA:

  • The sine of an angle is equal to the opposite side over the hypotenuse
  • The cosine of an angle is equal to the adjacent side over the hypotenuse
  • The tangent of an angle is equal to the opposite side over the adjacent side

If you have no idea what any of that is, review those relationships first and then dive into solving problems with angles of elevation and depression.

Example Problem 1

First of all, let's land a plane!

You're working in air traffic control and a plane is about to make an emergency landing at your airport. You're in charge of clearing out the part of the runway that the plane needs to land on. The angle of depression of the plane is 30 degrees, as shown in the picture. The plane is currently 10,000 feet above the ground and it's not quite over the runway yet. Will the plane make it to the runway before it hits the ground and if so, how many miles from the end of the runway will it touch down?

We'll start by labeling what we know. The problem tells us that the angle of depression is 30 degrees. In other words, if the pilot looked straight forward out of the cockpit window, his gaze would be tilted down 30 degrees below an imaginary horizontal line parallel to the ground.

We know that the plane is 10,000 feet up, so we can draw a triangle with the plane at one point.

We know that w is 90 degrees, which means that angle z must be 30 degrees. This fits with the rule that the angle of elevation and the angle of depression must be the same. If someone were unwisely standing on the ground where the plane is about to land and looking up at the plane, the angle of elevation of their gaze would be the same as the angle of depression of the pilot's gaze; in this case, they're both 30 degrees.

We can now use basic trigonometric relationships to solve for x and y because thanks to the TOA in SOH-CAH-TOA, we know that the tangent of 30 must equal the opposite side over the adjacent side, so tan(30) = 10,000/y.

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