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Angle Of Elevation And Depression

Shuvham Singhal
29/07/2017 0 0

Angles of Elevation and Depression:

You can use right triangles to findout  distances, if you know an angle of elevation or an angle of depression of a object. The figure below shows each of these kinds of angles.

Applications of Right Triangle Trigonometry Figure 1.svg

The angle of elevation is the angle between the horizontal line of sight and the line of sight up to an object. For example, if you are standing on the ground looking up at the top of a Chimeny , you could measure the angle of elevation. The angle of depression is the angle between the horizontal line of sight and the line of sight down to an object. For example, if you were standing on top of a hill or a building, looking down at an object, you could measure the angle of depression. You can measure these angles using a clinometer or a theodolite. People tend to use clinometers or theodolites to measure the height of trees and other tall objects. Here we will solve several problems involving these angles and

Q.1 You are standing 20 feet away from a tree, and you measure the angle of elevation to be 38°. How tall is the tree?


Solution:

The solution depends on your height, as you measure the angle of elevation from your line of sight. Assume that you are 5 feet tall. Then the figure below shows the triangle you are solving.

Applications of Right Triangle Trigonometry Example 3.svg

The figure shows us that once we find the value of T, we have to add 5 feet to this value to find the total height of the triangle. To find T, we should use the tangent value:

tan ? ( 38 ? ) = {\displaystyle \tan(38^{\circ })=\,\!} \tan(38^{\circ })=\,\! opposite adjacent = T 20 {\displaystyle {\frac {\text{opposite}}{\text{adjacent}}}={\frac {T}{20}}} {\frac {{\text{opposite}}}{{\text{adjacent}}}}={\frac {T}{20}}
tan ? ( 38 ? ) = {\displaystyle \tan(38^{\circ })=\,\!} \tan(38^{\circ })=\,\! T 20 {\displaystyle {\frac {T}{20}}} {\frac {T}{20}}
T = {\displaystyle T=\,\!} T=\,\! 20 tan ? ( 38 ? ) ≈ 15.63 {\displaystyle 20\tan(38^{\circ })\approx 15.63\,\!} 20\tan(38^{\circ })\approx 15.63\,\!
Height of tree ≈ {\displaystyle {\text{Height of tree}}\approx \,\!} {\text{Height of tree}}\approx \,\! 20.63   ft {\displaystyle 20.63\ {\text{ft}}\,\!} 20.63\ {\text{ft}}\,\!

The next example shows an angle of depression.

Q.2 You are standing on top of a building, looking at park in the distance. The angle of depression is 53°. If the building you are standing on is 100 feet tall, how far away is the park? Does your height matter?


Solution:

If we ignore the height of the person, we solve the following triangle:

Applications of Right Triangle Trigonometry Example 4a.svg

Given the angle of depression is 53°, angle A in the figure above is 37°. We can use the tangent function to find the distance from the building to the park:

tan ? ( 37 ? ) = {\displaystyle \tan(37^{\circ })=\,\!} \tan(37^{\circ })=\,\! opposite adjacent = d 100 {\displaystyle {\frac {\text{opposite}}{\text{adjacent}}}={\frac {d}{100}}} {\frac {{\text{opposite}}}{{\text{adjacent}}}}={\frac {d}{100}}
tan ? ( 37 ? ) = {\displaystyle \tan(37^{\circ })=\,\!} \tan(37^{\circ })=\,\! d 100 {\displaystyle {\frac {d}{100}}} {\frac {d}{100}}
d = {\displaystyle d=\,\!} d=\,\! 100 tan ? ( 37 ? ) ≈ 75.36   ft {\displaystyle 100\tan(37^{\circ })\approx 75.36\ {\text{ft}}\,\!} 100\tan(37^{\circ })\approx 75.36\ {\text{ft}}\,\!

If we take into account the height if the person, this will change the value of the adjacent side. For example, if the person is 5 feet tall, we have a different triangle:

Applications of Right Triangle Trigonometry Example 4b.svg
tan ? ( 37 ? ) = {\displaystyle \tan(37^{\circ })=\,\!} \tan(37^{\circ })=\,\! opposite adjacent = d 105 {\displaystyle {\frac {\text{opposite}}{\text{adjacent}}}={\frac {d}{105}}} {\frac {{\text{opposite}}}{{\text{adjacent}}}}={\frac {d}{105}}
tan ? ( 37 ? ) = {\displaystyle \tan(37^{\circ })=\,\!} \tan(37^{\circ })=\,\! d 105 {\displaystyle {\frac {d}{105}}} {\frac {d}{105}}
d = {\displaystyle d=\,\!} d=\,\! 105 tan ? ( 37 ? ) ≈ 79.12   ft {\displaystyle 105\tan(37^{\circ })\approx 79.12\ {\text{ft}}\,\!} 105\tan(37^{\circ })\approx 79.12\ {\text{ft}}\,\!

If you are only looking to estimate a distance, than you can ignore the height of the person taking the measurements. However, the height of the person will matter more in situations where the distances or lengths involved are smaller. For example, the height of the person will influence the result more in the tree height problem than in the building problem, as the tree is closer in height to the person than the building is.

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