 Hi, this is Chico. Let's continue on with our discussion of trigonometry, right? And what we've done so far is basically we did a little introduction to trig and we took a look at why it is that we study triangles and the main reason for us really delving really deep into triangles is because triangles allow us to analyze the perfect cyclic function because triangles are really related to circles, right? And cyclic functions are basically anything that repeats, right? Maybe the earth revolving around the sun, maybe the tides of the ocean, maybe just waves, right? Maybe sound, sound is a vibration, right? And that repeats, maybe light, maybe the female menstrual cycle. I will talk a little bit about all that stuff in the introductory video, right? And from there we went on and, you know, to analyze a circle we have to take a look at triangles, right? And what we did, we took a look at the three basic trig ratios, sine, cosine, and tangent and took a look at the unit circle and that basically allowed us to find out what happens to us as we move around the circle, right? What happens to our coordinate as we move around the circle and from there what we ended up doing, we basically graphed the three cyclic functions that we talked about, the three cyclic ratios, the three trig ratios, right? And we ended up graphing them on this side, right? And we had the sine function which is a perfect wave and the cos function which is also a perfect wave but it's sort of displaced and it's displaced 90 degrees, right? Because if we take this thing, the circle and rotate it 90 degrees, the x-axis goes up top, the y-axis comes down the bottom and basically we get the same thing as we have right now and the cos function is translated over, shifted over, it's called a phase shift over, right? And we ended up also graphing the tan function. So that's what we've done so far. Now we did all those videos based on the unit measure of an angle in degrees, right? And that's basically us moving around the circle like this and we called this angle degrees. We measured this in degrees, right? Which is something that we've been taught since we were really young, right? And measuring angles of something with a protractor or whatnot. Now what we're gonna do in this video is take a look at another type of way to measure angles and basically this is just another unit that we have, right? And it's called radians. And as I mentioned before, the reason that, you know, we study a unit circle is because the unit circle basically takes the radius and shrinks it down to one and the reason we work with one is because one is easily scalable, right? We can do anything we want with one, make it bigger, make it smaller, multiply by things. It's just an easy number to deal with, right? And that's one thing we do in mathematics is we try to simplify the world, right? We try to make things as easy as possible because while some of us I guess are lazy, right? That's sort of going a little extreme. But in mathematics what we like to do is simplify things as much as possible, right? If you're familiar with any type of mathematics or if you've looked at, you know, some of the some of the things we know in science, there are some amazing equations out there that have simplified complex systems in life into simple equations, sometimes two or three variables. And it's amazing to see these things where these equations basically represent something that has, you know, as human beings we we had no idea what what their inner workings were, right? And that's the core essence of mathematics. The core essence of mathematics is the language we use to analyze the world, right? It's a beautiful language and it's a simple language when you get into it, when you go into in-depth things, specific types of systems, the equations, the variables becomes complex, but throughout time what mathematicians have always tried to do is take those complex equations and combine them and see if all these variables are related into a single equation, right? And one of the ways we do this is we take any system that we're analyzing and we try to see how things are related, right? We try to see if there is a way for us to combine two variables that seem to be different into one variable that works for that system. And that's what radians is really about. It's about making the trig functions, the trig ratios for us to study a cyclic function simpler. And the way it does that, it takes the radius of a circle and it says, you know what? What we're going to do, we're going to measure the angle, right? Related, relative to the radius of the circle. And what it does is it says, let's say this is the radius and right now this is a unit circle so the radius is a 1, right? What we do with radians, we say let's take this the radius and make it equivalent to the arc length, okay? So take this 1 and rotate it and put it on here and if this is the radius is equal to 1, if you travel one unit around the arc length, around the perimeter, around the circumference of a circle, wherever you get to, we're going to call that one radian. So the way it works is if we're standing here, right? And this is our radius right now and we call this a unit circle. So our radius is equal to 1, right? Now this distance and one unit it could be one anything you want it to be, right? So if we travel around the arc length here, right? Around the circle the same distance, one unit. If this was one meter, if we travel around the circle one meter, then the angle we've gone to is equal to 1 radian. And for radians, you know, the unit what you put down is a rads, right? R-A-D. But usually when we're talking about in higher level mathematics and when you're talking about angles you're not really using degrees, you're using radians because radians relates the angle where you are on a circle directly to the radius. So it gets rid of one variable, degrees specifically, that is not related to anything on the circle, right? Which is not related to, well it is related to, it is related to the circle but it's not related to any other variable on the circle. And for a circle the most important variable, the most important thing about a circle is the radius. So if we can measure the angle based on the most important property of a circle, then we've gone one step, one huge step closer to simplifying the system. And for radians you can think about it as just another way of measuring angle, okay? So that's the way you should look at it. It's like using the imperial system to measure distance, you know, inches, feet, miles as compared to the metric system, you know, centimeters, meters and kilometers, right? But metric system is a much easier system to use an imperial system, that's why most of the world except I think three countries use the metric system because it's much simpler, it's multiples of 10, right? And it makes calculations easier, radians is the same thing, okay? So what we're going to do right now is draw another circle here and it doesn't necessarily have to be a unit circle, it's just going to be a circle. And what we're gonna do, we're gonna start doing our calculations in radians. And what we're gonna do, we're gonna see how conversions work going from radians to degrees. And it's really simple and it's really powerful and basically what we're gonna do is we're gonna do use ratios. And if you've been following the language of mathematics series four, that's what our focus is in series four, is doing unit conversions and talking about units and what units are and how to switch from one system to another system. And that's exactly what we're going to do here. And the way we're going to do it is we're gonna do ratios and cross multiplications and portionality. And that's the way I like teaching radians, right? A lot of teachers, a lot of places that I've seen, a lot of books that I've seen, the way they teach people to convert from degrees to radians is multiplying, you know, degrees by pi over 180 and multiplying radians by 180 over pi and getting, you know, doing the conversion that way. I personally don't like that, I actually really hate it because that requires you to memorize what ratio, what fraction you need to use to do the conversion. And I remember when I was learning this stuff, I really didn't like doing it that way because I could never remember which one it was if it was pi over 180 or 180 over pi. So what we're gonna do, we're gonna look at this in an intuitive type of way when it comes to doing the conversion. And this, what we're about to talk about, the way we're gonna do the conversions works for any system that you want to do unit conversions for. And it's extremely powerful. And coming out of high school, the most important thing you need to learn coming out of high school is how to do unit conversions and what fractions mean and what ratios mean. And because that basically allows you to dive into any system you want to dive into and do a conversion, switch from one system to another system if they're related, right? So it allows you to jump from one system to another system just by knowing, just by having a link between the two systems, right? It's proportionality, it's ratios, and all you need to do is do a cross multiplication and you've done your conversion. It's super powerful, okay? And that's what we're about to do right now when it comes to finding out how we can jump from degrees to radians, how we can do a unit conversion from degrees to radians, okay? So let's draw our circle again and start talking about what happens when we move around the circle, but not in degrees in radians. Okay, so what we're going to do is take this guy down. So where were we? We're about here. So what I'm going to do is I'm going to take my floss and put a little loop around it. So I'm going to slip this on here, right? Make that tight. And the trick for doing this, you have to hold the center tight and the string has to be tight or the floss has to be tight, okay? And what we're going to do, 9, 9, 1, 2, 3, 4, 5, 6, 7, 8, 9, 1, 2, 3, 4, 5, 6, 7, 8, 9, 1, 2, 3, 4, 5, 6, 7, 8, 9, 1, 2, 3, 4, 5, 6, 7, 8, 9. So let's do this. And that's our circle, right? Is it dark enough? It's dark enough, you can see that, right? So what we're going to do now is we're going to throw our grid on there, because we need our grid. And our grid, we basically, the reason, you know, one of the best ways to do mathematics, to analyze shapes is throw grids on things, right? So this is a circle standing in space somewhere. So what we end up doing is putting a grid system on it, that way we have coordinate system that we can, you know, move around the circle and go anywhere we want, right? So I'm going to bring my level, nice little level. It allows me to, with the bubble here, it allows me to make it straight, right? Not to mention, I have the grid system here. Let's put the x or the y up here. So this is our Cartesian coordinate system. This is our x-axis. And if you remember, we're going to call our x-axis, if this is going to be a unit circle, this is cos theta, right? And theta can be in degrees or radians. And this is going to be our y-axis and the y-axis is sin theta. So what we're going to do right now is, we're going to be standing here again. What we're going to do is going to move around the circle, right? And what we're going to do now is instead of measuring the angle as we move around the circle in degrees, we're going to measure it in radians. And the way this works is, our radius, this is our center of the circle. So our radius is r, agreed? And r can be anything. Since we're working with the unit circle, let's keep it as one. Okay. Because one thing is going to happen here right now is we're going to take a look at one of our magic numbers, right? So if our radius here is one, okay, what we're going to say is, let's move around the circle one, one unit around the arc length, right? So one distance is going to be approximately, let's see, it's going to be approximately here, okay? So what we've done is we moved around the circle the equivalent distance as the radius around the arc length. So we're up here now, right? And the distance that we've traveled here is, let's do this, the distance that we've traveled here is one unit. So we've gone from here to here. And the distance that we've gone is the same distance as the radius r. For our unit circle, it would be the same as one. So if this thing was five, we've traveled five this way. If this thing was 100, we've traveled 100 this way. And the way it works is the terminology we've come up with, what we say is if you travel around the arc length around the circle the same distance as the radius, then we're going to call this angle one radian. Pretty simple, right? So we're going to call this one radian. So this angle here, you can still call it theta, right? This theta here is going to be equal to one radian. It's as simple as that, right? Let's take a look at what happens as we continue to move around the circle. And what we're going to find out is something magic happens when we go halfway, okay? And it's a number that you've seen before. So if we travel two of these guys, we've gone here. If we travel three of these guys, we've gone here approximately, right? If we travel four, we've gone around here. If we travel five, we've gone around here. Actually, five would take us probably two around here, okay? I'm going to do a little quick calculation. I don't break it down very often. So five. So 360 is equal to... So what we're going to do is move around the circle based on integer values of radians. And the way it's going to look like is one radian is going to take us here, two radians is going to take us here, three radians is going to take us here, four radians is going to take us about here, five radians is going to take us here, six radians is going to take us here. And by the time we get to the end to where we were before, that's going to be 6.28 dot, dot, dot radians, okay? And the way this works is the connection we have between radians and degrees is something that you would have seen before, but you wouldn't have known where it comes from. If you remember, the circumference of a circle is 2 pi r, right? And 2 pi is how many radians it takes us to go around a full circle, okay? That's where the 2 pi comes from. So the link that we have between radians and degrees is if you go all the way around the circle, that's 2 pi radians. It means 2 times pi, which is pi is 3.14, you know, continues on dot, dot, dot. So in radian measures, a full circle is 6.28 dot, dot, dot, but we're not going to use dot, dot, dot. We're going to work in exact value. So we're going to call it pi. So in unit measure, movement around the circle to go where you were, the period of that circle is 2 pi. In degrees, that's 360 degrees. And that's the link, that's the connection we have between radians and degrees. And in your Language of Mathematics series 4, we've talked about this, right? If you have one system of measurement and another system of measurement, the way you can move from one system to another, all you need is a link between the two. Because as soon as you have a link between the two, you can do your unit conversions and take any number from this system converted to any number in this system, right? So what we're going to do right now is take a look at radians and see how we can convert them to degrees and vice versa. So we're going to see how we can jump between radians and degrees. And this is pretty important and it's really simple to do. So one rotation around the circle in radians is 2 pi. One rotation around the circle in degrees is 360 and that's our link. That's the connection that we have, right? So what we're going to do is we're going to do our calculations here. So we're going to convert from radians to degrees. The connection that we have is 2 pi radians is equal to 360 degrees. So as we talked about in series 4, all we got to do for this is as soon as we get a connection between these guys, we flip this thing either this way to get a ratio or we flip this thing this way to get a ratio. It's up to you. I'm going to go flip this guy this way, right? And what's going to happen is this ratio has to be proportional. It has to be equal to whatever the conversion we're going to do, right? So this is radians up top, degrees in the bottom, right? So we're going to go flip this thing. We're going to have 2 pi over 360, right? This connection, this ratio has to be proportional to radians divided by degrees, right? I put down radians up top and we got degrees in the bottom, right? So if I'm going to be staying consistent, I'm going to stay symmetrical and that's one thing that happens in mathematics. Mathematics is all about symmetry, right? So if I want to do a unit conversion, whatever angle I'm trying to find convert from radians to degrees or degrees to radians, they go here, right? So what I want to do right now, I want to find out what one radian is in degrees. So I'm going to go 1, right? Because that's my radian. I put my radians up top. I don't want to find out what one radian is in degrees. So I'm going to put my x here. And we've done a lot of this in series four and we're going to continue to do this in series four. So what we're going to do right now is we're going to cross multiply. We're going to take this guy, take it up there and take that guy, kick it up there, right? So what we've got is 2pi x is equal to 1 times 360, 1 times 360 is 360, right? I don't want to get x by myself. So I'm just going to do a division by 2pi, division by 2pi. So x is equal to 360 divided by 2pi. And for this stuff, we're going to need a calculator because they're not exact values when we do, when we try to do our calculations, when we convert from degrees to radians. So we've got our little calculator. We're just going to go 360 divided by 2, divided by pi. Comes out to 57.29578. Because we're dividing by an irrational number. So we're going to get an irrational number. So this is going to be 7.29578. And we're just going to round this up to 57.3 degrees, 57.3 degrees. So one radian is equal to 57.3 degrees. And that's what we've traveled here. So that is equal to 57.3 degrees. Two radians is going to take us to double this, right? So we're going to go, hold up our thing. So we're going to go 57.3 times 2, which is 114.6. So 114.6 takes us to here. So why not just list the stuff here, right? So two radians is equal to 114.6 degrees. We're going to go 57.3 times 3. Three radians is 171.9, 171.9 degrees. Four radians, four times 57.3 is 229.2 degrees. Five radians, 57.3 is equal to 286.5 degrees. Six radians is 343.8 degrees. And if we go seven radians, we've gone more than one rotation. We've gone past one cycle. Seven times 57.3 is 401.1 degrees. So what we're going to do right now is throw these angles, throw these lines on the circle approximately, right? So two radians is 114.6. And that's 90 degrees. So 114 takes you to about here. Three radians is 171.9, so 172. And that goes to about here. Does it look the same? Four radians is 229. So that comes out to about here. 225 takes you all the way in the center. So let's see, it's there. So five radians is 286. That's 270, so 286 is about here. 270 is 16 degrees. And six radians brings us to about here. 343. And if you go seven radians, as we've talked about with our terminology when it comes to coterminal reference and angle and standard position, if we go seven times 57.3, you get 401 degrees, right? If you subtract that from 360, you get 41 degrees. So seven radians brings you lower down here, right? So on the second rotation, you're less than 57 degrees. You end up here. And that's how many radians we have going around the circle, right? Now what's going to happen is if you travel, if we travel from here, from this point here, halfway around the circle to this point here, right? That ends up being pi radians, 3.14 dot, dot, dot, dot, dot. And that's where we get the number pi from. The distance from here, half a circle, is pi times whatever the radius was. So if this is one, the distance is 3.14, right? Dot, dot, dot. If this is 10, it's 31.4 dot, dot, dot. So that's where the magic number pi comes from. One of our unique numbers, one of our power numbers, one of the five that we have. Actually, I guess there's more, but the five that we deal with, zero, one, e pi, and the square root of negative one, those are the, I forget what they call it, the five magic numbers or whatever it is, right? That's one of the main numbers that comes into play in mathematics. So what we're going to do right now is do a few other unit conversions. Do a few other conversions between degrees to radians, and you'll see how powerful this is doing the conversion like this. Most schools, actually every school, every book that I've seen, does the unit conversion by saying, you know, if you want to convert from radians to degrees, you take whatever the radian measure is, and you multiply it by degrees over radians because they cancel each other out. Should we do this here? I'll show you how that works, but I'm not going to do it that way. I'm going to use the ratios because the ratios works for everything. And I like learning systems and I like learning methods of doing things that work everywhere, not memorizing, you know, ways to do it for one system and then having to memorize it to do it for another system. But I'll show that to you right now. So let's take this down. Hopefully the writing came out long distance, but if it didn't, this is what we had. So most places that I've seen, most teachers that I've seen, the way you do it is if you want to convert from radians to degrees. So you put whatever radian measure you had here, right? And what you want to do is you want to kill the radian, right? And if you remember how to multiply fractions, we've talked a lot about this. If you want to kill something up top here, you multiply it, you put the unit down here. So you multiply this thing by the link that we have for between radians and degrees. And the link that we have between radians and degrees is 360 degrees is the same as 2 pi radians. Now, we're going to simplify that a little bit and say if that's the case, 360 divided by 2 is 180. So if you go half a circle, right? The link that we have is 180 degrees is equivalent to pi. And that's the connection that we use, right? So if we want radians here, rad, we multiply it by, we want the rad down here, right? Because they're going to kill each other and we put the degrees up here. So in our case, if we want to find out what one radian is, we go one times the link that we have. You could go either 360 over 2 pi, but most places they teach it as 180 over 2 pi. So we multiply this by 180 over 2 pi, right? Which one times that is 180 divided by 2 pi. So 180, 180 divided by... Oh, not 180 divided by 2 pi. 180 divided by pi, right? So 180 divided by pi is 57.3 degrees, right? 57 point... Rounding up 57.3 degrees. And if you're going to convert from degrees to radians, to radians, what you do is you have your degrees up here. So you multiply this by radians over degrees. Because the degrees kills the degrees. So if we're going to convert anything, if we're going to go 57.3, convert it to radians, so let's do this thing. Let's go... We're going to make sure that it works the other way around, right? So we're going to go 57 degrees. We're going to multiply it by whatever our link is. And our link is 180 degrees is equal to 2, is equal to pi, right? So we want the degrees in the bottom. And we're going to have pi up here, okay? So if we do that, we're going to go pi divided by 180 is 0.0174. And we're going to multiply that by 57.3. And that gives us, you know, 1.000.0007.3, whatever it is. But it's really one, because this isn't an exact number, right? This is an irrational number, really, because we're dividing by an irrational number, multiplying by an irrational number. So this is equal to one radians. I'm not going to do any of the unit conversions that way. I'm not going to do any of the conversions between radians to degrees that way. I'm going to use what we talked about. The link that I have for full circle is 360 degrees is equal to 2 pi. So I'm going to go 360 degrees over 2 pi. This ratio has to be proportional to any degree that I have over any radians that I'm going to convert to or vice versa. So if I have a degree, I can convert it to radians because that'll become my ax, I put my degree there. If I have any radian, I can convert that to degrees because that'll be my number there, and that'll be my unknown there. So let's do a couple of these. So let's say we want to convert 115 degrees to radians. What we end up doing is convert. So the question would be convert 115 degrees to radians. So all we end up doing is going to use this ratio. So 360 degrees divided by 2 pi is equal to I have degrees and I want radians. So I'm going to put my degrees up here. 115 degrees divided by x, and that's the radians that I want. So again, what we're going to do is going to do cross multiplication. This guy goes up there. That guy goes up there. I'm going to have 360x is equal to 2 pi times 115. And then I'm going to divide by 360, divide by 360. So 360 killed 360. And again, we're going to do a lot of this and we have done some in series four of the language of mathematics, right? And we did some of the stuff in series three where we're doing cross multiplication where we're learning cross multiplication. But anyway, if we end up doing this we're going to go 2 times second function pi times 115 divided by 360. 2.0071 radians, right? Which is, you know, when we did it on the other page we found out that two radians was 114.6. So 115 degrees is 2.0x. Hopefully you can see this. I don't know if you can. It's 2.007 dot, dot, dot radians, right? Should we do a couple more? Just so you got this conversion down packed, right? So let's take this down and do a couple more conversions. So a few conversions. Let's make a table, maybe. Should we make a table? Here, we'll do two radians. Let's pick random numbers. Let's convert 7.5 radians to degrees. And let's convert negative 72 degrees to radians. And let's convert, I don't know, how about 130, actually 230. Sure. 213 radians to degrees. Because nothing says that, you know, we have to stay in these numbers, anything. You know, we don't have to be in one rotation, right? Let's go huge. Maybe we're looking to see what happens in multiple rotations of radians. And this is going to be huge in degrees, right? So again, our conversion is this. Let's do 7.5, right? So 360 degrees over 2 pi. That has to be equal to, I've given radians, I want degrees. So radians, I'm going to put that guy here. 7.5 x. And if you do your cross multiplication, I'm not going to do the full step of the cross multiplication, right? This guy just kicks up there, right? So x is equal to 7.5 times 360 divided by 2 pi. And again, you need calculator for this, right? So I'm going to go 7.5 times 360 divided by 2 divided by second function pi. So that's going to be 429.7 degrees. 429.7 degrees. Let's put this guy. Let's put it back in this. 400. Let's do this one. Sorry if I'm going a little fast right now, but I get excited when I do these conversions. I don't know why. It's just fun. The second one. I'm going to do the same thing, but instead of writing, I want radians out, right? So what I'm going to do is, I'm going to set up my conversion instead of 360 over 2 pi. I'm going to set it up as 2 pi over 360. The reason for that is, is because I want my x to be in the top. My unknown to be in the top. I'm looking for radians. I'm going to make sure the unknown is in the top. That way all I have to do is kick this up. I don't have to bring an x up the way we did on the previous one, right? So I'm going to go 2 pi over 360 degrees. That has to be equal to my unknown x over negative 72 degrees. And again, that guy is just going to kick up. So I won't bother writing this out. Let's write it out. x is equal to negative 72 times 2 pi divided by 360. Now, I know 2 goes into 360 180 times. So I can simplify, reduce the number of punches I have to do on a calculator by going 2 goes into 360 180 times. So I'm going to go 72 times second function pi divided by 180. I'm going to have 1.2566 radians. But this is negative, right? So negative is coming this way, right? So we're at 1.26 radians. So that distance there here. So we're about here, right? So this is going to be equal to negative 1.26 approximately, right? Let's do this huge one here. Over here, again I'm looking for I should let's put this one here. Negative 1.26 radians. For this one, I'm looking for degrees again. So I'm going to use this one because I'm going to have degrees up top. That way my x is up top. So I'm going to go 360 over 2 pi must be equal to x over 213. And all that happens is I'm going to kick up to 213. So again, 2 goes into 360 180 times, right? So 180 times 213 divided by second function pi is equal to 12,204 degrees. 12,204 degrees. Oops, I should write this down here. So x is equal to 12,204 degrees. Which makes sense because this number is going to be huge. The number of rotations around the circle is 2 pi. 2 times pi is 6.28. So 360. So what we can do just if you want to do an estimation, right? You can go 213 divided by 6. That's 35 more than 35.5 rotations. So if we can go 35.5 times 360 is 12,780. So I guess it was less than 35 rotations. So that's what we got right now. 213 times. That's how you convert from degrees to radians. And one of the reasons or the reason that I use proportionality ratios to do my unit conversions is is because this applies everywhere for any system that you want to convert to any other system. And I like doing things that way. That way I don't have to memorize things, right? 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 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, okay? And they're going to give you a certain radius and they're going to say what's the area, shaded area of part of a circle?