Lesson Explainer: Angles of Elevation and Depression Mathematics

In this explainer, we will learn how to solve real-world problems that involve angles of elevation and depression using the tangent ratio.

In order to solve problems involving angles of elevation or depression, we need to be able to determine which angle in the problem refers to an elevation or depression.

An angle of elevation refers to the angle made between an observer’s line of sight and a line horizontal to their eye when the object being observed is above the horizontal. This might be an airplane, the top of a tree, a balloon, or any other object that is above the horizontal line. It may also refer to the angle an object, such as a ladder, makes with the horizontal when leaning against a wall.

An angle of depression refers to the angle made between an observer’s line of sight and a line horizontal to their eye when the object being observed is below the horizontal. This might be an object on the ground if the observer is on a building or some other tall structure or the sea if the observer is on a cliff.

Definition: Angles of Elevation and Depression

An angle of elevation denotes the angle created by the line of sight of an observer and a horizontal line for an object above the horizontal.

An angle of depression denotes the angle created by the line of sight of an observer and a horizontal line for an object below the horizontal.

In the first example, we will identify which angle in a given diagram is the angle of elevation.

Example 1: Identifying the Angle of Elevation in a Diagram

In the given diagram of a ladder leaning against a wall, which of the following angles represents the ladder’s angle of elevation?

  1. ∠𝐡𝐴𝐢
  2. ∠𝐴𝐡𝐢
  3. ∠𝐴𝐢𝐡

Answer

The angle of elevation is the angle between the observer’s line of sight and the horizontal line (when the object is above the horizontal).

In this case, the angle of elevation is between the ladder and the horizontal (since there is no observer in this problem). This is highlighted in blue on the diagram below.

This angle is formed by the line segments 𝐴𝐡 and 𝐴𝐢 and is denoted as ∠𝐡𝐴𝐢 or ∠𝐢𝐴𝐡.

The answer is option A, ∠𝐡𝐴𝐢, since ∠𝐢𝐴𝐡 is not listed as an option.

Having learned how to identify the angle of elevation or depression within a diagram, we will next consider how to find unknown sides or angles in problems involving angles of elevation or depression.

In this explainer we will focus on problems that involve right triangles, where the line of sight of the observer, the horizontal line, and the perpendicular distance of the object being observed from the horizontal line form a right triangle. This can be seen in the cases of elevation and depression below.

By using trigonometry and the Pythagorean theorem, we can determine the lengths and angles of these right triangles when given either two lengths or a length and the angle of depression or elevation. Let’s recall the trigonometric ratios.

Definition: Trigonometric Ratios

For a right triangle with a non-right angle πœƒ, a hypotenuse of length 𝐻, a side opposite to πœƒ of length 𝑂, and a side adjacent to πœƒ of length 𝐴, sincostanπœƒ=𝑂𝐻,πœƒ=𝐴𝐻,πœƒ=𝑂𝐴.

For the right triangles we create in angles of elevation or depression, we can label the sides by using the angle as follows: the hypotenuse is the line of sight of the observer, the adjacent side is the horizontal line, and the opposite side is the perpendicular distance of the object being observed from the horizontal. We can see this in the diagram below.

Since problems involving angles of elevation or depression usually involve the distance of the horizontal line from the observer to the point above or below the object being observed (the adjacent side), the perpendicular distance from the object to the horizontal line (the opposite side), and the angle of elevation or depression, we use the ratio for these lengths. This is the tangent ratio.

Definition: Calculating Angles of Elevation and Depression

An angle πœƒ, which is the angle of elevation or depression formed by the line of sight of an observer and the horizontal line of an object above or below the horizontal, can be calculated using the following formula: tanπœƒ=𝑂𝐴, where 𝑂 is the length of the side opposite the angle of elevation or depression, or the perpendicular distance of the object from the horizontal line, and 𝐴 is the horizontal distance from the observer to the point above or below the object being observed.

In the next example, we will use the tangent ratio to calculate the angle of depression where a diagram is given.

Example 2: Calculating Angles of Depression

In the given diagram, Adam observes a buoy in the sea below him from a point 6 ft above a 45 ft cliff. He has been told that the perpendicular distance from the buoy to the base of the cliff is 60 ft. What is the angle of depression, in degrees, from Adam to the buoy? Give your solutions to two decimal places.

Answer

To solve this problem, we must start by labeling the diagram with the key information given in the question.

First, we are told that Adam observes a buoy in the sea below him from a point 6 ft above a 45 ft cliff. We can deduce that the total height of Adam and the cliff is the sum of 6 ft and 45 ft, which is 51 ft. On the diagram, this is represented by 𝐴𝐷. Since 𝐴𝐡𝐢𝐷 is a rectangle, then 𝐡𝐢 is the same length as 𝐴𝐷.

Next, Adam has been told that the perpendicular distance from the buoy to the base of the cliff is 60 ft. This is represented by 𝐷𝐢 on the diagram. Since 𝐴𝐡𝐢𝐷 is a rectangle, then 𝐴𝐡 is the same length as 𝐷𝐢.

We are required to find the angle of depression from Adam to the buoy. This is formed by Adam’s line of sight of the buoy, which is represented by 𝐴𝐢, and the horizontal line to Adam, which is 𝐴𝐡.

Since 𝐴𝐡 is perpendicular to 𝐡𝐢, then 𝐴𝐡𝐢 forms a right triangle. We can use the trigonometric ratios to find the angle of depression. Labeling the known sides gives us 𝐴𝐡 as the side adjacent to the angle and 𝐡𝐢 as the side opposite the angle.

Next, we need to use the trigonometric ratio for the opposite side and the adjacent side, which is the tangent ratio: tanoppositeadjacentπœƒ=, where πœƒ is the angle of depression.

By substituting 60 ft for the length of the adjacent side and 51 ft for the length of the opposite side, we get tanπœƒ=5160.

To solve for πœƒ, we must take the inverse tangent of the equation, giving us tanοŠ±οŠ§βˆ˜ο€Ό5160=40.364…, which to two decimal places is 40.36∘.

Therefore, the angle of depression from Adam to the buoy is 40.36∘ to 2 decimal places.

In the following example, we will again find the angle of elevation using the tangent ratio, but this time without a diagram.

Example 3: Using Trigonometry to Solve Word Problems Involving Angles of Elevation

Nabil stands 40 m from a building that is 25 m tall. What is the angle of elevation from Nabil to the top of the building? Round your answer to the nearest degree.

Answer

To solve this problem, we will first draw a diagram to represent the key information given. We are told that Nabil stands 40 m away from a building, which we can represent using a horizontal line. We are also told that the building is 25 m tall, which we can represent using a vertical line that joins to the horizontal line.

Since we are required to find the angle of elevation from Nabil to the top of the building, then we can draw a line from one end of the horizontal line to the top of the vertical line (also known as the line of sight).

Since the diagram shown is a right triangle, we can use the trigonometric ratios to find the angle of elevation. Labeling the known sides gives us the height of the building as the opposite side and the distance from Nabil to the building as the adjacent side.

Next, we need to use the trigonometric ratio for the opposite side and the adjacent side, which is the tangent ratio: tanoppositeadjacentπœƒ=, where πœƒ is the angle of elevation.

By substituting 40 m for the length of the adjacent side and 25 m for the length of the opposite side we get tanπœƒ=2540.

To solve for πœƒ, we must take inverse tangent of the equation, which gives tanοŠ±οŠ§βˆ˜ο€Ό2540=32.005…, which is 32∘ to the nearest degree.

Therefore, the angle of elevation from Nabil to the top of the building is 32∘ to the nearest degree.

In the next example, we will use the fact that the internal angles in a triangle sum to 180∘ in order to solve an angle of elevation problem involving a clinometer (an instrument used to measure the angle of elevation).

Example 4: Using Trigonometry to Solve Real-World Problems

Fares wants to find the height of a tower. He decides he needs to make a clinometer in order to measure the angle of elevation. He uses a straw, a protractor, some string, and a bit of Blu-Tack as a weight. Fares stands at a perpendicular distance of 100 ft from the base of the tower and measures the angle on his clinometer to be 59∘, as seen in the diagram.

  1. Work out the angle of elevation.
  2. Given that Fares’s eyeline is 6 ft from the ground, work out the height of the tower to the nearest foot.

Answer

Part 1

To find the angle of elevation, we first determine which angle this refers to on the diagram and then use the information given to us to calculate the angle.

By definition, the angle of elevation is the angle made by the line of sight of the observer and the horizontal line, where the object being observed is above the horizontal. In the case of the question, it is the line of sight from Fares to the top of the tower and the horizontal line from Fares to the tower. This is marked on the diagram.

As we are told, the angle measured by the clinometer is 59∘, and because the triangle formed by the line of sight, the horizontal line, and the perpendicular distance from the top of the tower to the horizontal line is a right triangle, we can use the fact that angles in a triangle sum to 180∘ to find the angle of elevation: angleofelevationangleofelevation+90+59=180=180βˆ’149=31.∘∘∘∘∘∘

Therefore, the angle of elevation is 31∘.

Part 2

To work out the height of the tower, we will first use the information from the question to label known values on the diagram. We are told that Fares stands a perpendicular distance of 100 ft from the tower. This is the same as the horizontal line from Fares’s eye to the point below the top of the tower, which is the base of the triangle in the diagram. We also know from part 1 that the angle of elevation is 31∘. We can label both of these on the diagram.

We are told later in the question that Fares is 6 ft tall. This is the same length as the distance from the base of the tower to where the horizontal line meets the tower. We are trying to find the total height of the tower, so we need to find the length from the horizontal line to the top of the tower, which we will call π‘₯. We will label both of these on the diagram as well.

Next, we will find π‘₯. Since we have a right triangle and know an angle and a side, we can use trigonometry to find the unknown side of length π‘₯. To do this, we will use the angle of elevation 31∘ as the known angle. This makes π‘₯ the length of the side opposite the angle and the horizontal line measuring 100 ft the length of the side adjacent to the angle, as seen in the diagram below.

Having labeled the necessary sides and angle, we can then use one of the trigonometric ratios to determine π‘₯. Since we know the length of the side adjacent to the angle and want to find the length of the opposite to the angle, we need to use the tangent ratio, which is tanoppositeadjacentπœƒ=, where πœƒ is the angle of elevation.

Substituting πœƒ=31∘, opposite=π‘₯, and adjacent=100, we have tan(31)=π‘₯100.∘

Solving for π‘₯, we get tanft(31)Γ—100=π‘₯,π‘₯=0.60086…×100β‰ˆ60.086.∘

Now that we have found π‘₯, we can find the height of the tower, which is the sum of π‘₯, 60.086 ft, and Fares’s height, 6 ft: heightofthetowerβ‰ˆ60.086+6β‰ˆ66.086, which is 66 ft to the nearest foot.

Therefore, the height of the tower is 66 ft to the nearest foot.

In the next example, we will consider a problem with a ladder leaning against a wall. This problem is slightly different to the previous 3 examples, as we are given different information and, as such, use a different trigonometric ratio.

Example 5: Using Right Triangle Trigonometry to Find an Unknown Length in a Real-World Context Involving Angles of Elevation

In the given diagram, a 15 ft ladder is leaning against a wall with an angle of elevation of 70∘. How high up the wall would it reach? Give your answer to two decimal places.

Answer

In order to find how high up the wall the ladder can reach, we need to first use the key information from the question to label the diagram. We know that the angle of elevation, which is the angle between the horizontal line 𝐴𝐢 and the ladder 𝐴𝐡, is 70∘. We also know that the length of the ladder 𝐴𝐡 is 15 ft. Since we are trying to find how high up the wall the ladder can reach, which is 𝐡𝐢, then we can call this π‘₯. Putting this on the diagram, we get the following.

Having put the key information on the diagram, we can see that we have one known side and one known angle in a right triangle. As such, we use trigonometry to find the unknown side of length π‘₯. To do this, we need to label the sides according the known angle πœƒ, which is 70∘. In this case, π‘₯ is the length of the side opposite the angle and the length of the ladder, 15 ft, is the length of the hypotenuse, as seen below.

As we know the angle and the hypotenuse and need to find the length of the side opposite the angle, then we use the sine ratio, which is sinoppositehypotenuseπœƒ=, where πœƒ is the angle of elevation.

By substituting 15 as the length of the hypotenuse, π‘₯ as the length of the opposite side, and 70∘ as angle πœƒ, we have sin(70)=π‘₯15.∘

Solving for π‘₯, we get sinft(70)Γ—15=π‘₯,π‘₯=0.9397…×15=14.095…,∘ which is 14.10 ft to two decimal places.

Therefore, the height the ladder can reach is 14.10 ft to two decimal places.

In this explainer, we have learned how to identify and find the angle of elevation or depression in problems and how to use trigonometric ratios to solve problems involving angles of elevation or depression.

Key Points

  • The angle of elevation or depression is the angle between the line of sight of an observer and the line horizontal to their eye when the object is above or below the horizontal.
  • The angle of elevation or depression, πœƒ, can be calculated using the following formula: tanπœƒ=𝑂𝐴, where 𝑂 is the length of the side opposite the angle of elevation or depression, or the perpendicular distance of the object from the horizontal line, and 𝐴 is the horizontal distance from the observer to the point above or below the object being observed.

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