Angles of elevation and angles of depression word problems are some of the most common problems in studies of trigonometry. These types of problems are famous for requiring the use of trigonometric ratios to relate a given angle and side length to find an unknown side length.
Whether it is the base of the cliff or the height of the lamp post, understanding how to solve angles of elevation and angles of depression word problems is an essential problem solving skill!
So let’s dig into it! It’s time to … elevate … your understanding!
Picture standing at the top of a tower and looking straight ahead along an imaginary horizontal line of sight. Suddenly, someone at ground level calls your name! You tilt your head downward from the imaginary horizontal line toward the ground through some angle.
As it turns out, this angle is the angle of depression! This angle is sometimes referred to as the angle of declination since there is a “decline” between the horizontal line of sight and the object being observed. An angle of depression must always be an acute angle.
Now, imagine that you are the person shouting up toward the top of the tower. You begin by looking straight ahead at the base of the tower, and tilt your head from ground level up toward the top of the tower.
The angle that you are tilting your head through is the angle of elevation. This angle is sometimes referred to as the angle of inclination since there is an “incline” between the horizontal line of sight and the top of the object. An angle of elevation must also always be an acute angle.
There are many famous angle theorems that can help us solve problems involving triangles. However, if you think back to your understanding of alternate interior angles, you should be able to see that these two angles are actually equal! This is because the line of sight and the ground are parallel lines.
Since we have a “Z-pattern”, we know that the angle of depression is equal to the angle of elevation! This is an important piece of knowledge that is going to help you solve any given angles of elevation or angles of depression word problems that you come across!
Before we take a look at the first example, watch this short video for an excellent visual representation of angles of elevation and angles of depression. I promise you won’t forget the difference after watching this video!
When it comes to solving angles of elevation and angles of depression word problems, the first step is to always start with a diagram to visualize the problem. Look for key phrases that give you hints about what specific angles and side lengths might be involved. Phrases such as the height of the first building give you an indication that that particular building will be important in your calculation.
From here, think about what trigonometric ratios will be involved. For example, the tangent ratio is used when you are relating the angle, the opposite, and the adjacent sides.
Let’s take a look at a few examples!
You are looking out a window at the very top of a tower. The height of the tower is known to be 50 feet. From the top of the building, you measure the angle of depression of a car to be 30 degrees. How far is the car from the building to the nearest foot?
To begin, we draw a diagram to help us visualize this scenario. The most important pieces of information given in this problem are the height of the tower (one of the sides of the right triangle) and the angle of depression of the car. Remember, since we are working with a building, we know that we have a right angle between the building and the ground.
The key trick to most angles of depression word problems is using your knowledge of alternate interior angles. This works since our line of sight and the ground are parallel lines!
We can use our understanding of alternate interior angles to show that the angle of depression from the top of the building is actually equal to the angle of elevation from the car up toward the tower!
Therefore, each of these angles are now known to be 30 degrees.
Looking at this new angle, we have the length of the side opposite to the angle (the height of the building) and we want to solve for length of the side adjacent to the angle (the distance from the car to the base of the tower). The trigonometric ratio relating the opposite and adjacent side lengths is the tangent ratio!
Setting up this ratio and solving for x results in:
Therefore, the car is 87 feet away from the base of the tower (rounded to the nearest foot).
You are standing on the ground and see two buildings in front of you that are the same distance away from you. You measure the angle of elevation to the top of the first building to be 45 degrees, and the angle of elevation to the top of the second building to be 60 degrees. You know that the height of the first building is 40 feet. What is the height of the second building to the nearest foot?
Just like the last problem, we start by sketching a diagram to help visualize this scenario. This scenario is slightly more complex since we are working with two buildings, so a diagram is important!
As it stands, the problem has not given us enough information to solve for the height of the second building, h. In order to solve the problem, we first need to determine the horizontal distance, d, between you and the base of the buildings. Once we have determined this horizontal distance, we can use it to solve for the height of the second building!
We use the height of the first building and the 45 degree angle (the angle of elevation) to set up a tangent ratio:
So we know the horizontal distance, d, is also 40 feet. From here, we can use the horizontal distance and the 60 degree angle of elevation to find the height of the second building!
$$ \begin tan60^\circ &= \frac \\ \\ 40 \times tan60^\circ &= h \\ \\ h &= 69.28 \\ \end $$
Therefore, the height of the second building is 69 feet!
Each angle of elevation or depression problem that you encounter will have similarities. You will always have to use the information given and a vertical line formed by some object to find the height of the tree, building, or tower.
If you aren’t give one, drawing a diagram should always be your first step. This is important to help you visualize the angles of elevation or depression that you are given. And as you have seen, having a good understanding of alternate interior angles and trigonometric ratios is also important!
For more practice on solving these types of problems, check out this angles elevation and angle of depression word problems worksheet!
My hope is that these examples have helped you understand the difference between angles of elevation and angles of depression. With this knowledge and a little practice, you will be able to solve any related problem that you come across!
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