 So what we're going to do right now, just to show how powerful this is, what we'll end up doing is we're going to do conversions, figure out what a certain area, segment areas, or a certain arc length is for a certain degree or for a certain radian. So let's take this down, let's come up with other types of questions that we can do unit conversion for using ratios. And one type of question you end up getting in mathematics is they're going to give you a circle, and they're going to give you a certain radius and they're going to say what's the shaded area of part of a circle. So let's do our calculations, let's do one problem here where you'll see how this works and how powerful this is. So let's draw a circle. Now I'm not going to put this on a grid, I'm just going to draw an arbitrary circle. Again I'm bringing out my little floss and putting this on it, so I'm just going to draw a random circle. So we have a circle, let's say our radius for this circle is 12. Hopefully you can see that, but let's say the radius is 12, so that's 12. And the question is going to be if we've traveled, let's say that's 110 degrees, 110 degrees, then what's the area of the shaded region here? How much have we covered? Now the connection, the ratio that we're going to use to do this conversion, to figure out this calculation is what we know about an area of a circle. The formula for area of a circle is pi r squared, right? Now I gave this degree measure, I gave the angle measure in degrees, right? So what we do is for your unit conversion, figuring out this calculation is we're going to use the same principle as we did with degrees over radians, but we're going to use area over the angle. So the conversion that you do is area of a circle is pi r squared. Pi r squared is for a whole circle, the area of a whole circle, right? If you go the whole circle, you're either going 360 degrees or 2 pi. This is the unit conversion we have. If you write this in equal notation and proportionality, right? You would have pi r squared equals to 360 degrees or 2 pi. I just wrote it as a fraction straight out. This ratio, this proportionality, this ratio has to be proportional to whatever we have here, right? So I'm just going to write it as general. It would be area divided by angle, super important. You end up using this a lot for calculations anyway. So let's see how this works for this, what our answer is going to be. This is the general ratio that we have, general unit conversion equation that we have. So whatever area you want to figure out, all you got to do is punch in the angle here. Conversely, if I gave you the area, you could figure out how many degrees or radians you've got. So let's do this calculation and we'll do another one with doing an area and figuring out what the angle is. So for this, my radius is 12, right? I gave the angle measurement in degrees, so I'm going to use 360, I'm not going to use 2 pi. If I gave the angle measurement in radians, you would have used 2 pi and not degrees, right? So I hope it makes sense when I put an or here, 360 or 2 pi, it really depends if you're doing your calculations in radians or degrees. So what we're going to do is we're going to go, for this calculation, we're going to go pi 12 squared divided by, we're in degrees, so I'm going to put 360 here, is equal to area, that's what I want to figure out, x over the area for what portion do I want, the portion that covers 110 degrees, right? And again, what you do is you cross multiply, you kick this up, that guy just goes up, right? So again, we need a calculator for this. So this is going to be pi times 12 squared, 12 squared is 144. So pi times 144 times 110 divided by 360. So the area, this area for the segment is 138.2. If this was meters, let's call this meters, so x is equal to 138.2 meters squared, if this was the radius, it was in meters, pretty powerful ratios. We're going to do one for arc length as well, but before we do that, let's do one where I've given you the area and you want to figure out the angle, right? So what we'll do is draw another circle, so let's put our circle dot here, then slip that out of my little knot here. So it's not going to be a beautiful circle for this one, but it's okay. We have a little slippage here. So let's say, again, the one thing that you do need for this is your radius, right? So let's do radius of, call this 25, okay? Let's say we've gone all the way down here, the radius is still 25, right? It's a circle, the radius is the same. Let's say we want to figure out the angle in radians, okay? And I'm going to say the area here, so let's say the area that I've given you here is 1407, right? Remember, 1407. So let's say the area I've given you here is 1407. So that's the area that we have here. And I want you to figure out the angle in radians. So what we end up doing is we're going to use the same ratio, the same equation, but instead of using 360, I'm going to use 2pi because in the question I would say, I want the answer to be in radians, okay? So what we have is, we're going to have pi r squared, which is 25 squared, divided by 2pi has to equal, I've given the area. We have the area. This is our unknown, the angle. So this is going to be 1407 over x. And the angle that we figure out is going to be in radians because we use 2pi, not degrees. So if we do our cross multiplication, this guy's going to kick up, that guy's going to go there, and then this guy's going to go down, but I'll do it in two steps so you see what's going on. So this is going to be pi times 250 squared x is equal to 2pi times 1407. And keep in mind, I could have killed the pi's right here, right? And then I want to get x by itself, so I'm going to divide by pi 25 squared. I'm going to divide by pi 25 squared, okay? And what I'm going to do, I'm going to kill the pi's right now. So what we end up having is 2 times 1407 divided by 25 squared, which is equal to 4.5 radians because that's what we're doing, right? We're doing our measurements in radians. So that ended up killing that. So x is equal to 4.5024, but we're not that accurate with this, right? Because we're losing information. Actually, we're not losing information because we killed the pi's. So 4.5024 radians, that's the angle we've gone. Pretty sweet, yeah? Let's do set up a same kind of ratio conversion, but instead of trying to figure out what an area is and how far what the angle is, we've traveled for a certain area, let's do it for arc length, right? The distance traveled. So let's draw another circle. Now what we'll do is I'm going to use the same angle as I did in the first one. That way there's a connection between the previous calculation and this calculation, right? So let's draw a circle again. So what we had was 110 degrees. So I'm going to draw this, and I'm going to go here, and I'm going to say this is 110 degrees, and this was 12, I believe. 12. So what I want to figure out now is what's the distance from here to here, right? What is that distance? So what we end up doing, the ratio, the equation we set up is, if you recall, the circumference of a circle is 2 pi r. So instead of having pi r squared up top and the angle in the bottom, we're going to have 2 pi r and the angles in the bottom. So the formula is 2 pi r over 360 degrees or 2 pi, depending if the angle you've been given is in degrees of radiance, right? And this is going to be equal to arc link divided by angle. So for this calculation, we want to find out what the arc link is when we travel 110 degrees for a circle of radius 12. So I'm going to use 2 pi r, that's a given, that's for sure we have to use that. The angle is in degrees, so I'm going to kill the 2 pi, I'm going to put 360 down here. I've been given the angle, I'm going to put 110 here, I'm going to figure out what the arc link is. So it's going to be 2 pi times 12 divided by 360 must equal the arc length over 110. And again, I'm just going to kick this up. So x is going to be equal to 2 times pi times 12 times 110. 2 times pi times 12 times 110. And I'm going to divide that by 360 divided by 360. So that's going to be 23.038234, whatever it is, right? Approximately 23 units, if this was meters, this is meters, right? If this was meters, that's meters. So we've traveled 23 meters around this circle. So let's draw another circle, and I'm going to give you the arc length, and we're going to figure out what the angle is we've traveled, right? Same as before. So we're going to set up our circle, and we're going to go... So we've got our circle set up again, and previously, in the last calculation, we said the radius was 25 and went all the way here, right? So instead of giving you the angle, I'm going to give you the arc length that we've traveled from there to there, right? So let's say, if this is 25, let's say we've traveled 112.5 around the circle for us to get here. So we've traveled, if this is in meters again, we've traveled 112.5 meters around the circle, and we want to figure out what our angle is, theta, in radians, right? Well, I'm going to use the same formula. 2 pi r, I want it in radians. I'm going to use 2 pi, not 360. I've given the arc length, I'm going to find the angle. So I'm going to go 2 pi times 25 divided by 2 pi has to be equal to 112.5 over x. And I'm going to do just to simplify the calculations. I want to kill the pi's, right? So x comes up, so this is going to be 2 times 25 is 50. Actually, I could kill the 2's as well. So 2's die as well. So this is going to be 25x is equal to 112.5. I'm going to divide by 25. I'm going to divide by 25. So x is going to be equal to 112.5 divided by 25 is 4.5 radians, 4.5 radians. And that's the angle that we've traveled on the circle to have an arc length of 112.5 meters if the radius of the circle was 25 meters. Learn these three equations. Understand them. You should know them intuitively. They should make sense. Right? As for what they were again, let's recreate them so you know. So if you're going to be converting between radians and degrees, you need to use this ratio. 360 over 2 pi is equal to degree radian. If you're going to try to do area, if the question involves areas, it's going to be pi r squared. Over 360 degrees or 2 pi is equal to area over, okay, super powerful. If the question is going to involve arc length, you're going to use 2 pi r over 360 degrees or 2 pi. It has to be equal to the arc length over the angle. And the angle depends, if it's in radians or degrees, or if you're given the arc length or whatnot, right? We did a couple of questions. These are the three equations that were ratios, proportionality, that you really have to know, be comfortable with. And you can do combinations of these, right? I could combine area and arc length. I could say pi r squared divided by 2 pi r. That's another proportionality ratio that I have, and that would have to equal the area over the arc length. And this ratios and proportionality and learning cross multiplication to do unit conversions, to jump from one system to another system is, again, it's super, super powerful. And you end up using it all over the place. And it is really understanding this concept of, you can compare any two things, any two systems, as long as you have a link in between. For our case for trigonometry right now, what we're doing is we're making a connection between radians and degrees. We're making a connection between area and an angle, maybe degrees or radians. We're making a connection between arc length and the angle, maybe degrees or radians, right? And that proportionality has to stay true for whatever circle that we're talking about, as long as we're not going outside of that system, right? Outside of that connection. For a circle right now, it basically means the radius of the circle has to stay constant, right? For whatever system that we're doing, for one calculation, right? So learn these three different equations, I guess, but these three concepts, and you can combine different ratios. We could have written down a fourth equation where we're comparing the area to the arc length. So we would have had pi r squared in the top and 2 pi r in the bottom. And over here, it would have been area and arc length, and this can go on forever. You can compare any two things, right? So that's how we do unit conversion. That's how we do converting between radians and degrees. And what we're going to do in the next video is take a look at some special triangles. And these special triangles are getting into the core part of where trigonometry comes into play, where a lot of different types of questions you end up getting in school regarding trig identities. And we end up getting some ratios that, you know, they get you to memorize. But for us, what we're going to do, we're going to generate a table. And these special triangles basically, just to give you a teaser, the way it works is, we take our unit circle and we take one quadrant, the distance from here to here, right? That's 90 degrees. And there are two special triangles we have. One of them is 45 by 45, or 45, 4590, which basically cuts the 90 degree angle in half, right? Because one thing that we like doing in mathematics, we like taking equal chunks out of something. We like looking at a system and taking chunks out, dividing things out evenly. So we take 90 degrees and cut it in half. That's one special triangle, which gives us 45 degrees. The other special triangle is, we take 90 degrees and cut it in three parts, right? And it gives us 60, 30, 60, 90 degrees and 60, 30, 90 degrees. It takes 90 degrees and breaks it up into 30 degrees segments, right? Useful. So that's what we're going to do in the next video. We're going to take a look at our special triangles and we're going to take a look at our trig ratios and find out what our trig ratios, what the ratios are for sine, cos and tan for our special triangles. And we're going to move around the unit circle. And we're going to take a look at those numbers, the ratios and see how they change. I'll see you guys in the next video. Bye for now.