In this video, we’re going to look at an application of trigonometry to calculating angles of elevation and depression.
So, first of all, let’s clarify what is meant by angles of elevation and angles of depression. And the two diagrams here are useful in explaining it, angles of elevation, first of all. So, we have here a diagram of a person and they are looking up at an object above them. The angle of elevation is the angle formed between the horizontal and that person’s line of sight as they’re looking up towards this object.
An angle of depression is a similar concept, but this time our observer is looking down at something. So, the angle of depression is the angle formed again between the horizontal and the observer’s line of sight as they look down towards an object.
In fact, if I were to add in another person, as I’ve done in this diagram on the right, then the angle of elevation from the observer below and the angle of depression from the observer above are the same. They’re congruent to each other because the two horizontals are parallel. And therefore, these two angles are alternate interior angles in parallel lines. So, a lot of questions about angles of elevation and depression tend to be worded questions, where you need to read a description, interpret it, and I would always suggest drawing a diagram to help. So, we’ll see some examples of this.
So, here is our first question. It says Tom stands on a cliff 100 metres high and sees a boat out at sea. The angle of depression from Tom to the boat is 51 degrees. Calculate the distance between the boat and the base of the cliff.
So, as I mentioned, a diagram is always a useful starting point to visualize the situation here. So, here is my diagram. I have a cliff with Tom on it and a boat out at sea and Tom is looking down towards the boat. Now, we’re told the angle of depression is 51 degrees. So, I also need to draw in the horizontal. Because, remember, the angle of depression is measured from the horizontal. So, there’s the horizontal. And the angle of depression we’re told is 51 degrees. So, I need to label that here.
Now, if I want to use trigonometry to approach this problem, then I need a right-angled triangle. So, I’m also going to add in a vertical line that is parallel to the cliff from the boat to the horizontal. So, this gives me the right-angled triangle that I’m going to work with. Now, we have some other information. We’re told that the cliff Tom standing on is 100 metres high. So that means I can label this measurement here as 100 metres.
Now, it’s just worth pointing out we haven’t accounted for Tom’s height in any way. Obviously, Tom is on top of the cliff, so he’s gonna be adding a little bit to this. But we haven’t been told how tall Tom is. So, in this question, we’re ignoring Tom’s height. In another question, it’s possible that you might be told how tall Tom is or how high up his eyes are from the ground, in which case you’d need to account for that as well. But in this question, we’re just treating it as the 100 metres for the height of the cliff.
Now, we’re asked to calculate the distance between the boat and the base of the cliff. So, that is this horizontal distance here, which I’ve labelled as 𝑑 metres. So, thinking about how to approach this problem then, it’s a trigonometry problem. And the first step for me when approaching a trigonometry question is always to label the sides as the opposite, the adjacent, and the hypotenuse in relation to the angle. So, that’s in relation to the 51 degrees here.
So, there are their labels. And I can see I’ve been given the opposite and I want to know the adjacent. So, I’ve got O and A. So, I’m going to be using the tangent ratio here. I need to recall then what the definition of the tangent ratio is. And the definition, remember, is that tan of 𝜃, where 𝜃 represents an angle, is the ratio of the opposite divided by the adjacent.
So, what I’m gonna do is I’m gonna write this ratio out again, but I’m gonna fill in the information that I know for this question. So, I’m gonna replace 𝜃 with 51 degrees, I’m gonna replace the opposite with 100, and I’m gonna replace adjacent with 𝑑 cause that’s the letter I’ve given it in this question. So, I have tan of 51 is equal to 100 over 𝑑.
Now, looking to solve this equation in order to work out the value of 𝑑. 𝑑 is currently in the denominator of a fraction, so the first thing I’m gonna do is I’m gonna multiply both sides of the equation by 𝑑. And this gives me 𝑑 multiplied by tan 51 is equal to 100. Now, the next step to getting 𝑑 on its own here is to divide both sides of the equation by tan 51. Tan 51 is just a number, so I can do that without any problems. And so, this gives me 𝑑 is equal to 100 over tan 51.
Now, at this point, I’m gonna reach for my calculator in order to evaluate that. And as that 51 was measured in degrees, I need to make sure my calculator is in degree mode for this question. So, evaluating that gives me 80.9784 for 𝑑. I haven’t been told how to round my answer. So, in this case, I’m gonna round it to the nearest metre. So, this tells me that the distance between the boat and the base of the cliff is 81 metres to the nearest metre.
So, really important to draw a diagram if you haven’t been given one. Read the information in the question carefully and make sure you put it onto the diagram properly. Recalling the tangent ratio and then solving the resulting equation in order to work out the length of the side we’re looking for.
Okay, the second question we’re going to look at. Jess stands 40 metres from a building 25 metres high. What is the angle of elevation from Jess to the top of the building?
So, as I suggested before, a diagram would be a really good place to start with this question. So, this time, I’ve just represented Jess by a dot. We haven’t been told anything about her height. So, we’re not taking that into account. She’s 40 metres from the building, which is 25 metres high. And we are making the reasonable assumption here that the building is at a right angle to the floor, which is horizontal. So, we’re looking to calculate the angle of elevation as Jess looks up at the top of the building. So, it’s this angle that I’ve marked as 𝜃 on the diagram.
So, as before, it’s a trigonometry problem, so always sensible to label the three sides of the triangle first. So, the hypotenuse, the longest side here, the opposite which is the side opposite that angle 𝜃, and then the adjacent which is between 𝜃 and the right angle. The two sides we’ve been given are the opposite and the adjacent. So, we’re going to be using that tangent ratio again. So, let’s write down its definition. So, tan of 𝜃 is equal to the opposite divided by the adjacent.
So, what I’m gonna do is I’m gonna write this ratio down. But I’m gonna replace the opposite and the adjacent with their values in this question, so 25 and 40. So, I have tan of 𝜃 is equal to 25 over 40. Now, that does actually simplify as I can divide both parts of this ratio through by five. So, if I wanted to, I could simplify it to five over eight. Now, as we’re looking to calculate an angle this time rather than a side, we need to use that inverse tan function in order to do this.
So, I have that 𝜃 is equal to tan inverse of five over eight. And at this point, I’m gonna reach for my calculator to evaluate that. So, this tells me that 𝜃 is equal to 32.00538. And if I round that to the nearest degree then, I have my answer to this question, which is that the angle of elevation is 32 degrees. So, as before, start with a diagram, identify the sides of the right-angled triangle, and then use the tan ratio in order to find this missing angle.
Okay, the final question that we’re going to look at says, Sue stands four and a half metres away from a statue. The angle of elevation from Sue to the base of the statue is 18 degrees. The angle of elevation from Sue to the top of the statue is 49 degrees. We are asked to work out how tall the statue is.
So, just think about the information in that question carefully for a moment. We’re given two angles of elevation. And one of those is to the base of the statue, which means this statue isn’t flat on the floor with Sue. It’s above her in some way. So, we need to take that into account when we draw our diagram. So, the situation looks something like this. We have Sue standing on the ground and she’s looking up towards the statue. I’ll draw in the horizontal for the ground as well.
So, now let’s add in the information that we know. Sue is standing four and a half metres away. So, the horizontal distance here is four and a half metres. We’re also given two angles of elevation. The angle of elevation from Sue to the base is 18 degrees. So, that’s this angle here. And the angle of elevation from Sue to the top is 49 degrees. So, that’s this whole angle here, measured from the horizontal. And what we’re looking to work out is the height of the statue. So, we’re looking to work out this distance here, which I’ll refer to as 𝑥 metres.
Now, that length 𝑥 is not actually in a right-angled triangle. And we need right-angled triangles in order to do this type of trigonometry. So, it’s going to be a two-stage process in order to work out this length 𝑥. There are two right-angled triangles in the diagram. So, what I’m going to do is draw them out separately so that we can visualize the problem a little more easily.
So, here are those two right-angled triangles. And let’s just match them up with the diagram. This length here, which I’m going to call 𝑦, is this length here in the original diagram. So, that’s the smaller right-angled triangle at the bottom. This length here in the larger triangle, so this one here which I’m going to call 𝑧, this is this total length here in the diagram. So perhaps, you can see now what my strategy is going to be. I’m going to use these two right-angled triangles to calculate 𝑦 and 𝑧. And then looking at the diagram, you can see that 𝑥 will be the difference between these two values. So, 𝑥 will be 𝑧 subtract 𝑦.
So first of all, in each of these triangles, I’m gonna label the three sides, the hypotenuse, the opposite, and the adjacent relative to these angles of 18 and 49 degrees. So, in both cases, we know the adjacent and we want to work out the opposite, which means I’m gonna be using the tan ratio. So, I’ll recall the definition of the tangent ratio. And remember, it’s this, that tan of 𝜃 the angle is equal to the opposite divided by the adjacent.
So, I’m going to apply this twice. Starting with the smaller triangle then, I’m gonna replace 𝜃 with 18 degrees and I’m gonna replace the adjacent with 4.5. So, I have tan of 18 is equal to 𝑦 over 4.5. The first step to solving this equation then is I need to multiply both sides of the equation by 4.5 because that’s currently in the denominator. So, this gives me that 𝑦 is equal to 4.5 tan 18. And I’ve just written the two sides of the equation the other way round there. Now, if I evaluate that on my calculator, it’s a value of 1.46. But I’m actually gonna keep it exact for now cause I need to use 𝑦 again later in the question. And if I keep it like this, then it will be an exact value and I won’t introduce any rounding errors.
So, now, I look at the second triangle. And again, I want to write out the tangent ratio but filling in the information I know. So, 𝜃 is going to be replaced with 49 and adjacent is again going to be replaced with 4.5. So, I have tan of 49 is equal to 𝑧 over 4.5. As in the previous triangle, I now need to multiply both sides of this equation by 4.5. And so, I have that 𝑧 is equal to 4.5 tan 49. Now, again, if I were to evaluate that, it’s a value of about 5.18. But I’m gonna keep it like this for now.
Right, the final step is I wanted to work out the height of this statue, which was 𝑥. And remember we said that in order to do that, I would need to do 𝑧 minus 𝑦. So, I have 𝑥 is equal to 4.5 tan 49 minus 4.5 tan 18. And this is the first stage, where I’m gonna use my calculator to evaluate that. So, this tells me that 𝑥 is equal to 3.7145. Now, this answer is in metres. So, if I round it to the nearest hundredth, that would be the nearest centimetre. So, this tells me that the height of the statue then is 3.71 metres.
So, in this question, getting a diagram set up correctly at the beginning was really important. Don’t assume that the statue is standing on the same flat ground as Sue. We had to read the question carefully and deduce it is actually above her.
In summary then, we’ve defined what angles of elevation and depression are. And we’ve seen how to use the tangent ratio in order to answer questions involving calculating angle of elevation or depression or calculating a missing length from a worded description.