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.

Before you start with this explainer, you should be confident finding angle measures and missing sides using right triangle trigonometry and the laws of sines and cosines.

Now, prior to looking at examples and recalling trigonometric ratios and the laws of sines and cosines, we will define what angles of elevation and depression are.

Definition: Angles of Elevation and Depression

An angle of elevation is the β€œupward” angle from the horizontal to a line of sight from the object to a given point, whereas an angle of depression is where the angle goes β€œdownward” from the horizontal to a given point, as shown below.

Now, we will look at the steps for solving a problem involving angles of elevation or depression where a right triangle can be formed.

  • First, where one is not already drawn in the question, sketch a right triangle to represent the given scenario.
  • Next, label all the known distances and the given angle of elevation/depression.
  • Finally, use the trigonometric functions sine, cosine, and tangent to find the unknown distances or angles.

In our first example, we will solve a problem where a diagram has already been drawn.

Example 1: Finding an Unknown Length given a Real-World Problem and a Diagram

The distance between two buildings is 40 m. The top of building 𝐢𝐷 has an angle of elevation of 30∘, measured from the top of building 𝐴𝐡. If the height of building 𝐴𝐡=30m and the bases of the two buildings are on the same horizontal plane, then the height of 𝐢𝐷 to the nearest metre=m.

Answer

In the problem, we can see that the scenario has been represented in a diagram, and, from the information given, a right triangle has been formed. It is this triangle that we will look at first.

In order to find the height of 𝐢𝐷, we will need to calculate π‘₯. This is because if we add π‘₯ to the height of 𝐴𝐡, then this will give us the height of 𝐢𝐷.

We are going to use the trigonometric ratios to helps us find π‘₯. To use trigonometric ratios with a right triangle, you can follow these four steps:

  1. Label the sides.
  2. Choose the trigonometric ratio.
  3. Substitute in known values.
  4. Solve.

Let’s apply this to our problem.

Step 1

First, we label the hypotenuse, the opposite, and the adjacent.

Step 2

To choose the correct ratio, we look at the side we have been given and the one that we want to find. In our problem, these are the opposite and the adjacent. Using this information, we select our ratio. There is a memory aid we can use in order to assist with the choice.

This memory aid shows us which sides we require for each ratio. As we have the opposite (O) and the adjacent (A), we will use the tangent ratio. This tells us that tanoppositeadjacentπœƒ=.

Step 3

We now substitute in our known values, tan30=π‘₯40.∘

Step 4

Finally, we rearrange and solve for π‘₯: π‘₯=40(30).tan∘

To maintain accuracy, we will leave π‘₯ in this form until the final calculation.

Now, to find the height of 𝐢𝐷, we add π‘₯ to the height of 𝐴𝐡, which we are told in the question is 30 m: 𝐢𝐷=40(30)+30𝐢𝐷=53.094….tan∘

In the question we are asked to give the number of metres to the nearest metre; therefore, the answer is 53 metres.

In our next example, we will explore a problem that can once again be solved using right triangle trigonometry. However, in this example, we will need to draw a diagram to represent the scenario and we will also be looking at both an angle of depression and an angle of elevation.

Example 2: Using Right Triangle Trigonometry to Solve Word Problems Involving Angles of Elevation and Depression

A building is 8 metres tall. The angle of elevation from the top of the building to the top of a tree is 44∘ and the angle of depression from the top of the building to the base of the tree is 58∘. Find the distance between the base of the building and the base of the tree giving the answer to two decimal places.

Answer

The first step in solving this problem is to draw a sketch of the problem that is labeled with all the information that we have been given.

In the sketch, we have labeled the distance that we are trying to find as π‘₯. Now, from this sketch, we can extract a right triangle to help us calculate this distance.

As we now have a right triangle and are looking for a missing side, we can use trigonometric ratios to find π‘₯.

First, we label the hypotenuse, the opposite, and the adjacent.

Now, to choose the correct ratio, we look at the side we have been given and the one that we want to find. In our problem, these are the opposite and the adjacent. Using this information, we select our ratio: tanoppositeadjacentπœƒ=.

Next, we substitute in the known values and then solve to find π‘₯: tantantan58=8π‘₯π‘₯58=8π‘₯=858π‘₯=4.9989….∘∘∘

Finally, as asked for in the question, we round to two decimal places to give the distance between the base of the building and the base of the tree: 5.00 m.

In the next question, again we use right triangle trigonometry; however, in this example, we will need to use problem-solving skills to find the distance required.

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

The height of a lighthouse is 60 metres. The angles of elevation between two boats in the sea and the top of the lighthouse are 29∘ and 39∘ respectively. Given that the two boats and the base of the lighthouse are colinear and that the boats are both on the same side of the lighthouse, find the distance between the two boats giving the answer to the nearest metre.

Answer

The first step in solving this problem is to draw a sketch of the problem labeled with all the information that we have been given.

Now, as we are told that the two boats and the base of the lighthouse are colinear, we can create two right triangles that can enable us to find the distance between the two boats, which we have labeled 𝑑.

The next step is to use trigonometric ratios to calculate the distance between each boat and the lighthouse. Once we have these distances, we can calculate the difference, and this will give us the distance between the two boats.

We will begin with the boat that is further away. First, we label the hypotenuse, the opposite, and the adjacent.

Now, to choose the correct ratio, we look at the side we have been given and the one that we want to find. In our problem, these are the opposite and the adjacent. Using this information, we select our ratio: tanoppositeadjacentπœƒ=.

Next, we substitute in the known values and then solve to find π‘₯ (the distance from the lighthouse): tantantan29=60π‘₯π‘₯29=60π‘₯=6029.∘∘∘

At this stage, we will leave π‘₯ in this form to maintain accuracy. Now, we will find the distance of the boat closer to the lighthouse in the same way:

tantantan39=60𝑦𝑦39=60𝑦=6039.∘∘∘

We now have the distances of both boats from the lighthouse; therefore, we can calculate the distance between them as 𝑑=π‘₯βˆ’π‘¦.

If we subtract 𝑦 from π‘₯, we get 6029βˆ’6039=34.1490….tantan∘∘

Therefor, we can say that the distance between the two boats, to the nearest metre, is 34 m.

In our next example, we are going to look at a problem where we are given two angles of depression and we need to use the sine rule to find the solution. Before we start, let us quickly recall the sine rule.

Formula: The Sine Rule

Consider the given triangle.

We have π‘Ž(𝐴)=𝑏(𝐡)=𝑐(𝐢).sinsinsin

Example 4: Finding an Unknown Height by Forming and Solving a System of Equations

A tower is 33 metres tall. The angle of depression from the top of a hill to the top of the tower is 31∘. The angle of depression from the top of the hill to the bottom of the tower is 52∘. Find the height of the hill given the bases of the hill and the tower lie on the same horizontal level. Give the answer to the nearest metre.

Answer

To help us solve this problem, we are going to draw a sketch of the problem labeled with all the information that we have been given. We have also labeled some points with 𝐴, 𝐡, 𝐢, and 𝐷.

From the diagram, we can see that what we are trying to find is the length of 𝐡𝐢. In order to do this, we need to find the missing length: π‘₯.

Since 𝐴𝐡 is horizontal and 𝐡𝐢 is vertical, we know that ∠𝐴𝐡𝐢=90∘. Now, looking at triangle 𝐴𝐡𝐢, we can see that we have two of the angles, so ∠𝐴𝐢𝐡=180βˆ’52βˆ’90=38.∘

Also, we have that ∠𝐷𝐴𝐢=52βˆ’31=21.∘

Let us now focus on triangle 𝐴𝐷𝐢.

We can find the length of 𝐴𝐷 using the sine rule. The sine rule tell us that 𝐴𝐷(𝐢)=𝐷𝐢(𝐴).sinsin

Substituting in what we know and rearranging, we can find the length of 𝐴𝐷: 𝐴𝐷=33(38)(21)β‰ˆ56.693.sinsinm∘∘

Now, let us look at triangle 𝐴𝐡𝐷.

Since this is a right triangle, we can apply the trigonometric ratio sinOpp.Hyp.(πœƒ)=. Using πœƒ=31∘, Hyp.=56.693, and Opp.=π‘₯, we find the value of π‘₯ to be π‘₯=56.693(31)β‰ˆ29.199.sinm∘

Finally, to find the height of the hill, we need to add π‘₯ to 33

π‘₯+33=62.198873….

Therefore, we can say that the height of the hill to the nearest metre is 62 m.

In our final example, we will demonstrate how to apply this procedure to a function involving the natural logarithm.

Example 5: Using Angles of Elevation to Solve a Real-World Problem

The angle of elevation of the top of a hill from its base is 37∘. A man climbs the hill from that point at an angle of 26∘ to the horizontal for a distance of 340 metres. His path then continues to the top at an angle of 69∘ to the horizontal. Find the height of the hill to the nearest metre.

Answer

Let us start by drawing a sketch of what this question describes.

In this diagram, we can see that the base of the hill where the man starts climbing is at 𝐴, the top of the hill is at 𝐡, the height of the hill is β„Ž, and the point where the angle of the path changes is at 𝐢.

By looking at the angles at 𝐴, we can see that ∠𝐢𝐴𝐡=11∘. Next, let us consider the triangle outlined in green below.

Since the angles in this triangle must sum to 180∘, we have that ∠𝐢𝐡𝐷=21∘. Similarly, if we consider triangle 𝐴𝐡𝐷, we can see that ∠𝐴𝐡𝐷=53.∘

Using the two angles we have found at 𝐡, we are now able to say that ∠𝐴𝐡𝐢=32.∘

Next, we will consider triangle 𝐴𝐡𝐢.

Since we already know two angles in this triangle, we find that ∠𝐴𝐢𝐡=137∘. We are now able to apply the sine rule to this triangle to find the length of 𝐴𝐡. In doing this, we find 𝐴𝐡(137)=340(32)𝐴𝐡=340(137)(32)β‰ˆ437.575.sinsinsinsinm∘∘∘∘

Finally, if we consider the right triangle 𝐴𝐡𝐷, we will be able to find β„Ž using trigonometric ratios.

From the diagram, we can see that sin(37)=β„Ž437.575.∘

Rearranging this for β„Ž, we get our solution, which is that the height of the hill is β„Ž=263m to the nearest metre.

We will finish by recapping the key points from this explainer.

Key Points

  • An angle of elevation is the β€œupward” angle from the horizontal to a line of sight from the object to a given point, whereas an angle of depression is where the angle goes β€œdownward” from the horizontal to a given point.
  • Drawing a sketch of the given scenario is often a good place to start when solving problems of angles of elevation and depression.
  • We can use the trigonometric ratios to solve simple problems of angles of elevation and depression.
  • Sometimes, we will require the law of sines or the law of cosines in order to solve problems of angles of elevation and depression.

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