What are Angles of Elevation And Depression?

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There are times in both mathematics and real life where it's helpful to know an object's location compared to a fixed point. If that fixed point is on the horizon or some other horizontal line, this may require you to calculate the angle of elevation or angle of depression for the object. If this sounds confusing, don't worry. These angles are just references to where an object or point is located above or below that horizon.

TL;DR (Too Long; Didn't Read)

Angles of elevation and depression are angles that rise (elevation) or fall (depression) from a point on a horizontal line. Calculate them by assuming a right triangle and using sine, cosine or tangent.

What Is an Angle of Elevation?

The angle of elevation of a point or object is the angle at which you would draw a line to intersect the point from a single point (often referred to as the "observer") on a horizontal line. If you were to pick a point on the x-axis of a grid and draw a line from that point to another point somewhere above the x-axis, the angle of that line in comparison to the x-axis itself would be the angle of elevation. In a real-world scenario, the angle of elevation could be viewed as the angle you would look at compared to the ground around you when you look up into the sky to see a bird flying.

What Is an Angle of Depression?

In contrast to the angle of elevation, the angle of depression is the angle at which you would draw a line from a point on a horizontal line to intersect another point that falls below the line. Using the x-axis example from before, the angle of depression would require you to choose a point on the x-axis and draw a line from it to another point that was somewhere below the x-axis. The angle of that line in comparison to the x-axis itself would be the angle of depression. In the bird scenario, imagine the bird itself flying along an imaginary horizontal plane. The angle that the bird would look along to look down and see you standing on the ground would be the angle of depression.

Calculating the Angles

To calculate the angle of elevation or angle of depression for an object from any point on a horizontal line, assume that the observer and the point or object being observed make up the two non-right corners of a right triangle. The hypotenuse of the triangle is the line drawn between the two points (observer and observed), and the right angle of the triangle is created by drawing a vertical line from the observed point to the horizontal line the observer stands on. Calculate the angle for the corner marked by the observer, using the height of the observed object (in comparison to the horizontal line the observer is on) and its distance from the observer (measured along the horizontal line) to make the calculation. With the height and distance, you can use the Pythagorean Theorem (a2 + b2 = c2) to calculate the hypotenuse of the triangle.

Once you have the height, distance and hypotenuse, use sine, cosine or tangent as follows:

\sin(x) = \frac{\text{height}}{\text{hypotenuse}}
\cos(x) = \frac{\text{distance}}{\text{hypotenuse}}
\tan (x) = \frac{\text{height}}{\text{distance}}

This will give you the ratio of the two sides you selected. From here, you can calculate the angle by using the inverse function of the function you chose to generate the initial ratio (sin-1, cos-1 or tan-1). Enter the appropriate inverse function (and your ratio from before) into a calculator to get your angle (θ), as seen here:

\sin^{-1}(x) = θ \\ \cos^{-1}(x) = θ \\ \tan^{-1}(x) = θ

Point/Observer Congruence

In most cases, you can assume that the angles of elevation and depression between a point or object and its observer are congruent. Both the point and its observer exist on horizontal lines that are assumed to be parallel. As a result, the angle at which you look up at a bird would be the same angle at which it looks down at you, if measured against parallel horizontal lines originating at you and the bird. This does not hold when line curvature or radial orbits are taken into account, however.

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