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Prentice Hall Pre-Algebra: Online Textbook Help13 chapters | 135 lessons

Instructor:
*Betsy Chesnutt*

Betsy teaches college physics, biology, and engineering and has a Ph.D. in Biomedical Engineering

Angles of elevation and depression can be used to find unknown heights and distances. In this lesson, you can practice using angles of elevation and depression to solve problems.

Imagine you are walking through a forest and look up at the top of a tall tree. The angle between the horizontal ground and your line of sight to the top of the tree is known as the **angle of elevation**. Similarly, if you are looking down at something below you, the **angle of depression** is measured between the horizontal and your line of sight downward to the object.

You can use angles of depression and elevation, along with the trigonometric functions sine, cosine, and tangent to calculate unknown distances. Let's look at three practice problems that will help you understand how to do this.

*There is a tall tree in your backyard and you think it might hit your house if it fell over. You measure that the base of the tree is 48 ft from your house, but you don't know how tall the tree is. To determine the height of the tree, you stand just outside your back door and measure the angle of elevation from the ground to the tree to be 64 degrees. How tall is the tree?*

To figure this out, first carefully draw a picture of the situation and label all the distances and angles that you know. In this case, you know the distance to the base of the tree (48 ft) and the angle of elevation from the ground to the tree (64 degrees).

Then, you will use one or more of the trig functions to find the missing side of the triangle (the height of the tree, *h*). Here, since you know the angle and the adjacent side and you want to find the opposite side of the triangle, you will want to use the tangent function:

So, you now know that the tree is 98 ft tall. Would it fall on your house or not? Possibly, yes! If the tree fell toward your house, it would certainly hit because the tree is 98 ft tall and there are only 48 ft from your house to the base of the tree.

*Standing on a 35 m high cliff, you look down on your friend who is standing on the flat ground in front of the cliff. The angle of depression along the line of sight from you to your friend is 65 degrees. How far away from the cliff is your friend?*

First, notice that to find the angle inside the triangle, you will need to subtract the angle of depression from 90 degrees.

90 degrees - 65 degrees = 25 degrees

Even though this problem may initially sound very different from the first practice problem, it is really very similar. Once you draw a picture of the situation and label all your known distances and angles, you should see that you can use the tangent function to find the unknown distance once again.

Now, let's try to also find the straight line distance from you to your friend. For this, you will need to use another of the trigonometric functions, sine.

Let's call the distance from you to your friend *x* and then calculate *x* using the sine function:

Now let's try one that is a little more complicated:

*A building is on a 50 m tall hill. If you stand 70 m from the hill and look up at the building, the angle of elevation to the bottom of the building is 20 degrees and the angle of elevation to the top of the building is 60 degrees. How tall is the building?*

Let's first try to make this problem look a little simpler by pulling out the relevant information. The most important thing to notice is the angle of elevation to the top of the building (60 degrees). Then, you can see that the height of the right triangle is the height of the building plus the height of the hill (*h*+50 m).

Then, you can once again use the tangent function to find the height of the building.

The angle between the horizontal ground and your line of sight to the top of the tree is known as the **angle of elevation**. Similarly, if you are looking down at something below you, the **angle of depression** is measured between the horizontal and your line of sight downward to the object.

To solve problems involving angle of depression or elevation, first carefully draw a right triangle and label all the known distances and the angle of depression or elevation. Then, use the trigonometric functions sine, cosine, and tangent to to find the unknown distances.

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8 in chapter 11 of the course:

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Prentice Hall Pre-Algebra: Online Textbook Help13 chapters | 135 lessons

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