WEBVTT mathematics/geometry/pyo
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Welcome back to Educator.com.
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For this next lesson, we are going to go over a couple of different types of angles: angles of elevation and angles of depression.
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The first one, angle of elevation: now, we know that "to elevate" means to go up.
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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.
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Any angle that is formed by an increasing line, like that, with a horizontal line, is called an angle of elevation.
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This angle right here is the angle of elevation.
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Now, keep in mind that it has to be with a horizontal line; we can't have a vertical line with an increasing line.
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Even though this is increasing, this looks like the angle of elevation, but it is actually not.
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This is not considered an angle of elevation; it has to be a horizontal line and an increasing line.
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That is the angle of elevation; the next one is angle of depression.
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We know that "to be depressed" or "to depress" something is to go down.
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The angle of depression is an angle that is formed by, again, a horizontal line, and a decreasing line.
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It has to go downwards; and that is the angle of depression.
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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.
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It has to be with a horizontal line and a decreasing line.
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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.
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But just make sure that one of the sides of the angles is horizontal; the other one has to be going downwards.
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Straight into our examples: Name the angle of elevation and the angle of depression.
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Here we have several different angles: we have angle ABC; we have angle BCD; we have angle BDC; we have angle CBD--
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all of these different angles, and we have to name which one is the angle of elevation and the angle of depression.
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We know that an angle of elevation is one that is going up; it is horizontal and going up.
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This is horizontal; that thing is going up; here is another horizontal, and going up.
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So then, this angle right here would be the angle of elevation; the angle of elevation is angle BCD.
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Or, for this one, you can just say angle C.
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This is considered the angle of elevation.
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The angle of depression, one that is horizontal and going down, is going to be this one right here, the angle of depression.
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Now, this one cannot be the angle of depression; angle DBC, even though it looks like it is going down,
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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.
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Then, this cannot be an angle of depression; it would just be angle ABC; you can also say angle CBA.
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Can you say angle B?--no, you can't for this one, because there are different angles formed at that vertex, with this B.
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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.
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But with this one, that is not the case; it has to be angle ABC or angle CBA.
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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.
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A ski slope is 650 yards long with a vertical drop of 200 yards; find the angle of depression of the slope.
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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.
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A vertical drop: that means from here, all the way, just vertical.
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So, let's say this is flat ground; here, a vertical drop is going to be 200 yards.
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Here is a right angle; find the angle of depression of the slope.
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The angle of depression: now, is that the angle?--no, because this is a vertical line.
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Is this the angle of depression? No, that would be the angle of elevation.
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So, what I have to do--let me erase this person here, and let me also erase this and write it lower...
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this is 650 yards...I am going to form my own angle of depression.
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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,
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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.
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It is the angle formed using the slope as one of the sides.
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So, we can just draw a fake line and have the angle that is formed with it; we can use that.
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Or the angle of elevation would be this angle right here, using the slope.
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This is going to be x; now, here we have 200, and we have 650.
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Now, even though they say "angle of depression," I know that this angle of depression and this angle of elevation are the same angle.
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And how do I know that? Here, this is a horizontal line and a horizontal line.
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If they are both horizontal, that means that they are both parallel.
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If they are both parallel, and this is a transversal, then alternate interior angles are congruent.
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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.
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So then, we know that this angle with this angle is congruent, because they are parallel, and these are alternate interior angles.
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And then, why would I want to use this angle instead of that angle--why did I transfer this variable to this angle?
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It is because I have a triangle here; and when I have a right triangle, then I have so many different options.
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But if it is there, then what am I going to do with that?
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Or I can also...if that is kind of difficult for you to see, then what you can do is:
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from here, you can also draw another fake line, going down, like that (that is not so vertical; I'll draw it again).
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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.
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So, either way, you can either transfer this to there, or you can just draw this and then transfer this to there.
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The same thing: it does not matter, as long as you just use the right triangle.
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Using this right here, I have an angle, and what else do I have?
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To find this, I have side opposite, and I have hypotenuse.
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Now, I know I have to use trigonometric functions; I have to use those ratios, because again,
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an angle with a side--whenever you are using angles with sides, then you have to use those trigonometric functions,
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especially when the angle is what you are looking for.
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Which one do I use? Let me write it again: Soh-cah-toa.
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Which one uses opposite and hypotenuse? Right here are opposite and hypotenuse; that would be sine.
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Here is that; this is a different formula; this is a different formula.
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We are going to use the first formula: sine of x is equal to opposite (is 200), over the hypotenuse (is 650).
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So, from here, I need to find x--I need to find the angle measure.
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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.
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So, if the angle measure is the variable--that is what you are looking for--then you have to let the calculator know.
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If you punch in sine of this number, then your calculator is going to think that this is the angle measure.
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Make sure you use 2nd, sin, so that then you have that sin^-1...200 divided by 650; close the parentheses.
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And you are going to get that x is equal to 17.9 degrees; that is the value of x.
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Now, if you are still not understanding the trigonometric functions of this--
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why you have to find the inverse sine function here--it is because,
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if I have an angle measure, sine of, let's say, 50 degrees; you have to have an angle measure next to the sine.
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That means that automatically you are going to punch in the sine button and 50.
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And then, that is going to give you the answer.
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See how this is x; the calculator is going to give you x, and then what is it? x is .7660.
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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.
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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.
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You have to make sure; you have to punch in inverse sine, because that way you are letting the calculator know
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that this number is not the angle measure; this is the answer, and I want the angle measure.
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So, you are kind of doing the opposite.
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So then, on your calculator, you are going to punch in that inverse sine.
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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.
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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.
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How high above the ground is the kite?
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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).
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And the length of the string right here is 40 feet long, and it makes a 35-degree angle with the ground.
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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.
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This is x: how high is it above the ground?
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Here, whenever you have a word problem like this, make sure you draw it out.
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You don't have to draw these pictures, but make sure that you draw out your triangle.
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Then, we have to figure out how we are going to find the missing side or angle.
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Now, here we are given this angle; so from that angle's point of view...
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And I know that I am going to use Soh-cah-toa, because again, we have angles with sides.
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Here, from this angle's point of view, I have opposite, and I have hypotenuse.
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Which one am I going to use? According to Soh-cah-toa, I have to use "oh," that one.
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That means the sine of 35 degrees, because the ratio then becomes "sin(x) = opposite/hypotenuse."
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The sine of 35 is equal to...the side opposite is x, over the hypotenuse, which is 40.
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Then, I take my calculator; I am going to punch in sin(35), and that is .5736.
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That is equal to x/40; how do I solve for x?
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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.
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You are going to get 22.94; and that is x, right there.
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So then, the kite is going to be 22.94 feet above the ground.
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And let's go over the last example: here, it is just that they are giving you the sides.
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This right here, the measure of angle A, is 35; and this, we know, is the angle of elevation.
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So, if they said that the angle of elevation is 35, then it would be the same thing here, angle A.
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AB is 9, and BC is x; so again, here we are looking at the distance from B to C.
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We are going to use, from this angle's point of view, the opposite and the hypotenuse, this one right here.
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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.
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So then, go ahead and try to solve it: sine of 35 is .5736; that equals x/9.
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And I am going to multiply this side by 9 and multiply this side by 9.
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So, my answer is that x is 5.16; this BC (and I can just say BC instead of x) is 5.16.
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Just make sure to keep in mind that trigonometric functions are these three: Soh-cah-toa is made up of three trigonometric ratios.
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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.
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These are three formulas to keep in mind.
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And that is it for this lesson; thank you for watching Educator.com.