### 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

An angle of elevation is formed by a horizontal line and an increasing line.

An angle of depression is formed by a vertical line and an decreasing line.

Angle of depression: ∠CAD

m∠C = 35

^{o}, AB = 4, find BC.

- tan C = [AB/BC]
- tan35
^{o}= [4/BC] - 0.7 = [4/BC]

Both the angle of elevation and the angle of depression contain a horizontal line.

.

- sin x = [300/500] = 0.6
- x = sin
^{ − 1}0.6

^{o}.

AC = 6, AB = 4, find m∠A.

- cos A = [AB/AC] = [4/6] = [2/3]
- m∠A = cos
^{ − 1}[2/3]

^{o}

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.

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.

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

### Geometry Online Course

### Transcription: Angles of Elevation and Depression

*Welcome back to Educator.com.*0000

*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 650...now, 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 2 ^{nd}, 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 know*0797

*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 2^{nd}, 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 Educator.com.*1261

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