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Angles of Elevation and Depression

  • Angle of Elevation: Angle formed by a horizontal line and an increasing line
  • Angle of Depression: Angle formed by a horizontal line and a decreasing line

Angles of Elevation and Depression

Determine whether the following statement is true or false.
An angle of elevation is formed by a horizontal line and an increasing line.
Determine whether the following statement is true or false.
An angle of depression is formed by a vertical line and an decreasing line.
Name the angle of elevation and depression.
Angle of elevation: ∠BAC
Angle of depression: ∠CAD

m∠C = 35o, AB = 4, find BC.
  • tan C = [AB/BC]
  • tan35o = [4/BC]
  • 0.7 = [4/BC]
BC = 5.714.
Determine whether the following statement is true or false.
Both the angle of elevation and the angle of depression contain a horizontal line.
Determine whether ABD is an angle of depression.
David is walking from the bottom of a hill to the top. He walked 500 meters, the hill is 300 meters high, find the angle of elevation of the slope.
  • sin x = [300/500] = 0.6
  • x = sin − 1 0.6
x = 38.9o.

AC = 6, AB = 4, find m∠A.
  • cos A = [AB/AC] = [4/6] = [2/3]
  • m∠A = cos − 1 [2/3]
m∠A = 48.2o
Determine whether the following statement is true or false,
Given the lengths of any two sides of a right triangle, we can find the length of the third side and the measures of the two acute angles.
Determine whether the following statement is true or false,
Given the measures of the two acute angles of a right triangle, we can find the lengths of three sides.

*These practice questions are only helpful when you work on them offline on a piece of paper and then use the solution steps function to check your answer.


Angles of Elevation and Depression

Lecture Slides are screen-captured images of important points in the lecture. Students can download and print out these lecture slide images to do practice problems as well as take notes while watching the lecture.

  • Intro 0:00
  • Angle of Elevation 0:10
    • Definition of Angle of Elevation & Example
  • Angle of Depression 1:19
    • Definition of Angle of Depression & Example
  • Extra Example 1: Name the Angle of Elevation and Depression 2:22
  • Extra Example 2: Word Problem & Angle of Depression 4:41
  • Extra Example 3: Word Problem & Angle of Elevation 14:02
  • Extra Example 4: Find the Missing Measure 18:10

Transcription: Angles of Elevation and Depression

Welcome back to

For this next lesson, we are going to go over a couple of different types of angles: angles of elevation and angles of depression.0002

The first one, angle of elevation: now, we know that "to elevate" means to go up.0012

So, an angle of elevation would be an angle, like this, where it is formed by a horizontal line, right there, and an increasing line.0020

Any angle that is formed by an increasing line, like that, with a horizontal line, is called an angle of elevation.0031

This angle right here is the angle of elevation.0044

Now, keep in mind that it has to be with a horizontal line; we can't have a vertical line with an increasing line.0050

Even though this is increasing, this looks like the angle of elevation, but it is actually not.0063

This is not considered an angle of elevation; it has to be a horizontal line and an increasing line.0069

That is the angle of elevation; the next one is angle of depression.0075

We know that "to be depressed" or "to depress" something is to go down.0083

The angle of depression is an angle that is formed by, again, a horizontal line, and a decreasing line.0094

It has to go downwards; and that is the angle of depression.0101

Again, it has to be a horizontal line; if it is a vertical line with a decreasing line, this right here is not an angle of depression.0107

It has to be with a horizontal line and a decreasing line.0121

And it could go either way; it can go this way, like this; it could go that way; this is still an angle of depression.0124

But just make sure that one of the sides of the angles is horizontal; the other one has to be going downwards.0133

Straight into our examples: Name the angle of elevation and the angle of depression.0144

Here we have several different angles: we have angle ABC; we have angle BCD; we have angle BDC; we have angle CBD--0150

all of these different angles, and we have to name which one is the angle of elevation and the angle of depression.0161

We know that an angle of elevation is one that is going up; it is horizontal and going up.0168

This is horizontal; that thing is going up; here is another horizontal, and going up.0174

So then, this angle right here would be the angle of elevation; the angle of elevation is angle BCD.0181

Or, for this one, you can just say angle C.0199

This is considered the angle of elevation.0203

The angle of depression, one that is horizontal and going down, is going to be this one right here, the angle of depression.0206

Now, this one cannot be the angle of depression; angle DBC, even though it looks like it is going down,0224

is going down; it is just not called an angle of depression, because it has to be with a horizontal line; this is a vertical line.0232

Then, this cannot be an angle of depression; it would just be angle ABC; you can also say angle CBA.0241

Can you say angle B?--no, you can't for this one, because there are different angles formed at that vertex, with this B.0255

So, with this one, you can, because there is only one angle; if I say angle C, then you know exactly what angle I am talking about.0264

But with this one, that is not the case; it has to be angle ABC or angle CBA.0271

The next one: here is where we are actually going to be using those terms, "angle of depression" and "angle of elevation": it is mostly with word problems.0282

A ski slope is 650 yards long with a vertical drop of 200 yards; find the angle of depression of the slope.0292

A ski slope is, you know that a ski slope goes like this; we have a person skiing...this whole thing is 650 yards.0304

A vertical drop: that means from here, all the way, just vertical.0326

So, let's say this is flat ground; here, a vertical drop is going to be 200 yards.0330

Here is a right angle; find the angle of depression of the slope.0340

The angle of depression: now, is that the angle?--no, because this is a vertical line.0351

Is this the angle of depression? No, that would be the angle of elevation.0356

So, what I have to do--let me erase this person here, and let me also erase this and write it lower...0360

this is 650 yards...I am going to form my own angle of depression.0372

I am going to make a horizontal line here; and then, I know that this angle right here is going to be the angle of depression,0383

because it is the angle of depression of the slope: this is the slope, so if we are on the slope, the angle of depression would be this.0395

It is the angle formed using the slope as one of the sides.0405

So, we can just draw a fake line and have the angle that is formed with it; we can use that.0410

Or the angle of elevation would be this angle right here, using the slope.0420

This is going to be x; now, here we have 200, and we have 650.0427

Now, even though they say "angle of depression," I know that this angle of depression and this angle of elevation are the same angle.0437

And how do I know that? Here, this is a horizontal line and a horizontal line.0451

If they are both horizontal, that means that they are both parallel.0458

If they are both parallel, and this is a transversal, then alternate interior angles are congruent.0463

So, if that is x, then this is also x; and to draw that out, here is this line; here is the ground; and here is that slope, the transversal.0469

So then, we know that this angle with this angle is congruent, because they are parallel, and these are alternate interior angles.0487

And then, why would I want to use this angle instead of that angle--why did I transfer this variable to this angle?0496

It is because I have a triangle here; and when I have a right triangle, then I have so many different options.0504

But if it is there, then what am I going to do with that?0510

Or I can also...if that is kind of difficult for you to see, then what you can do is:0512

from here, you can also draw another fake line, going down, like that (that is not so vertical; I'll draw it again).0525

And then, what you can do is just transfer this 200; if that is 200, this has to be 200; and then, you have a triangle here.0539

So, either way, you can either transfer this to there, or you can just draw this and then transfer this to there.0547

The same thing: it does not matter, as long as you just use the right triangle.0558

Using this right here, I have an angle, and what else do I have?0565

To find this, I have side opposite, and I have hypotenuse.0581

Now, I know I have to use trigonometric functions; I have to use those ratios, because again,0595

an angle with a side--whenever you are using angles with sides, then you have to use those trigonometric functions,0601

especially when the angle is what you are looking for.0607

Which one do I use? Let me write it again: Soh-cah-toa.0612

Which one uses opposite and hypotenuse? Right here are opposite and hypotenuse; that would be sine.0621

Here is that; this is a different formula; this is a different formula.0629

We are going to use the first formula: sine of x is equal to opposite (is 200), over the hypotenuse (is 650).0635

So, from here, I need to find x--I need to find the angle measure.0657

You go to your calculator, and make sure you don't punch in sine of this number, because you can only find sine of angle measures.0662

So, if the angle measure is the variable--that is what you are looking for--then you have to let the calculator know.0677

If you punch in sine of this number, then your calculator is going to think that this is the angle measure.0683

Make sure you use 2nd, sin, so that then you have that sin-1...200 divided by 650; close the parentheses.0690

And you are going to get that x is equal to 17.9 degrees; that is the value of x.0703

Now, if you are still not understanding the trigonometric functions of this--0720

why you have to find the inverse sine function here--it is because,0726

if I have an angle measure, sine of, let's say, 50 degrees; you have to have an angle measure next to the sine.0734

That means that automatically you are going to punch in the sine button and 50.0753

And then, that is going to give you the answer.0758

See how this is x; the calculator is going to give you x, and then what is it? x is .7660.0764

If you punch in sine of 50, the calculator knows that that 50 is an angle measure, and then it is going to give you x.0778

In this case, we don't have the angle measure; x is what the calculator gives you, so you want the calculator to give you the angle measure.0784

You have to make sure; you have to punch in inverse sine, because that way you are letting the calculator know0797

that this number is not the angle measure; this is the answer, and I want the angle measure.0803

So, you are kind of doing the opposite.0811

So then, on your calculator, you are going to punch in that inverse sine.0813

It is right above the sin button, sin-1; you are probably going to have to push 2nd, and then sin, and then that number.0820

The next problem: Susanna is flying a kite; the length of the string is 40 feet long, and it makes a 35-degree angle with the ground.0842

How high above the ground is the kite?0859

Let's say that this is Susanna, and she is flying a kite; the kite is up like that, and the kite, let's say, is right here (I'll make this shorter).0864

And the length of the string right here is 40 feet long, and it makes a 35-degree angle with the ground.0898

So then, the ground is 35 (it is so hard to see)...and they want to know how high above the ground the kite is.0910

This is x: how high is it above the ground?0936

Here, whenever you have a word problem like this, make sure you draw it out.0943

You don't have to draw these pictures, but make sure that you draw out your triangle.0953

Then, we have to figure out how we are going to find the missing side or angle.0959

Now, here we are given this angle; so from that angle's point of view...0965

And I know that I am going to use Soh-cah-toa, because again, we have angles with sides.0969

Here, from this angle's point of view, I have opposite, and I have hypotenuse.0975

Which one am I going to use? According to Soh-cah-toa, I have to use "oh," that one.0985

That means the sine of 35 degrees, because the ratio then becomes "sin(x) = opposite/hypotenuse."0993

The sine of 35 is equal to...the side opposite is x, over the hypotenuse, which is 40.1008

Then, I take my calculator; I am going to punch in sin(35), and that is .5736.1018

That is equal to x/40; how do I solve for x?1044

I have to multiply it by 40; then I take this and multiply by 40; without clearing the calculator, you can just go ahead and multiply by 40.1048

You are going to get 22.94; and that is x, right there.1062

So then, the kite is going to be 22.94 feet above the ground.1072

And let's go over the last example: here, it is just that they are giving you the sides.1090

This right here, the measure of angle A, is 35; and this, we know, is the angle of elevation.1102

So, if they said that the angle of elevation is 35, then it would be the same thing here, angle A.1109

AB is 9, and BC is x; so again, here we are looking at the distance from B to C.1116

We are going to use, from this angle's point of view, the opposite and the hypotenuse, this one right here.1139

And that is going to be the sine of 35, the same angle as the last problem, equals x (because it is opposite) over hypotenuse, which is 9.1148

So then, go ahead and try to solve it: sine of 35 is .5736; that equals x/9.1174

And I am going to multiply this side by 9 and multiply this side by 9.1188

So, my answer is that x is 5.16; this BC (and I can just say BC instead of x) is 5.16.1203

Just make sure to keep in mind that trigonometric functions are these three: Soh-cah-toa is made up of three trigonometric ratios.1223

The sine of x equals opposite over hypotenuse; the cosine of x equals adjacent over hypotenuse; and tangent of x is equal to opposite over adjacent.1236

These are three formulas to keep in mind.1255

And that is it for this lesson; thank you for watching