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Video: Trigonometry: Angles of Elevation and Depression

Lauren McNaughten

Learn the definition of angles of elevation and depression. Use your knowledge of the tangent ratio to calculate angles of elevation and depression or distances in a range of word problems.

14:04

Video Transcript

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 a hundred meters high and sees a boat out at sea. The angle of depression from Tom to the boat is fifty-one 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 fifty-one 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 fifty-one 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 angle triangle that I’m going to work with now. Now we have some other information. We’re told that the cliff Tom standing on is one hundred meters high. So that means I can label this measurement here as one hundred meters.

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 need to account for that as well. But in this question, we’re just treating it as a hundred meters 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 meters. 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 fifty-one 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 the 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 fifty-one degrees, I’m gonna replace the opposite with a hundred, 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 fifty-one is equal to one hundred over 𝑑. Now looking to solve this equation in order to work out the value of 𝑑, 𝑑 is coming 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 fifty-one is equal to a hundred. Now the next step to getting 𝑑 on its own here is to divide both sides of the equation by tan fifty-one. Tan fifty-one is just a number, so I can do that without any problems. And so this gives me 𝑑 is equal to one hundred over tan fifty-one.

Now at this point, I’m gonna reach for my calculator in order to evaluate that. And as that fifty-one was measured in degrees, I need to make sure my calculator is in degree mode for this question. So evaluating that gives me eighty point nine seven eight four for 𝑑. I haven’t been told how to round my answer, so in this case I’m gonna round it to the nearest meter. So this tells me that the distance between the boat and the base of the cliff is eighty-one meters to the nearest meter. 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 gonna look at: Jess stands forty meters from a building twenty-five meters 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 forty meters from the building, which is twenty-five meters 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 gonna 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 going to 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 twenty-five and forty. So I have tan of 𝜃 is equal to twenty-five over forty. Now that does actually simplify as I can divide both parts of this ratio, three 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 was 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 thirty-two point zero zero five three eight. And if I round that to the nearest degree, then I have my answer to this question, which is at the angle of elevation is thirty-two 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 gonna look at says Sue stands four and a half meters away from a statue. The angle of elevation from Sue to the base of the statue is eighteen degrees. The angle of elevation from Sue to the top of the statue is forty-nine 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 on 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 meters away, so the horizontal distance here is four and a half meters. Well once we were given two angles of elevation, the angle of elevation from Sue to the base is eighteen degrees. So that’s this angle here. And the angle of elevation from Sue to the top is forty-nine degrees, so that’s this whole angle here measured from the horizontal. And what we’re looking to workout is the height of the statue, so we’re looking to work out this distance here which are referred to as 𝑥 meters. Now that length 𝑥 is not actually in a right-angled triangle. And we need right angle 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 eighteen and forty-nine 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 will call the definition of the tangent ratio. And remember it’s this, that tan of 𝜃 to the angle is equal to the opposite divided by the adjacent. So I’m gonna apply this twice starting with the smaller triangle then. I’m gonna replace 𝜃 with eighteen degrees and I’m gonna replace the adjacent with four point five.

So I have tan of eighteen is equal to 𝑦 over four point five. The first step to solving this equation then is I need to multiply both sides of the equation by four point five, because that’s currently in the denominator. So this gives me that 𝑦 is equal to four point five tan eighteen. 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 one point four six. But I’m actually gonna keep it exact for now, because 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 by filling in the information I know. So 𝜃 is gonna be replaced with forty-nine and adjacent is again gonna be replaced with four point five. So I have tan of forty-nine is equal to 𝑧 over four point five. As in the previous triangle, I now need to multiply both sides of this equation by four point five. And so I have that 𝑧 is equal to four point five tan forty-nine. Now again, if I were to evaluate that, it’s a value of about five point one eight. But I’m gonna keep it like this for now.

Right. The final step is I wanted to work out the height of the 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 four point five tan forty-nine minus four point five tan eighteen. 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 three point seven one four five. Now this answer is in meters. So if I round it to the nearest hundred, that would be the nearest centimeter. So this tells me that the height of the statue then is three point seven one meters.

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’s actually above her. In summary then, we’ve defined what the 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 of depression or calculating a missing length from a word of description.