 So what we're going to do is we're going to take these down and create another circle. So what we'll do is we'll keep this page up so we ignore terminology, but when we do need to use that space and we will need to use that space, we'll take that off, right? And what we're going to do right now is recreate our circle, but we're going to make a unit circle, which basically means that we're going to set the radius equal to 1. And the reason that we do this is because in mathematics we like to simplify our calculations, right? And that's that's the crazy about the aspect about mathematics that people find math hard, is they don't have an appreciation for the fact that math is really the language we use to optimize the world, to optimize our understanding of the world. So what math is really about is simplifying things, crunching what's happening in the real world, in the world we interact with, into simpler and simpler equations and simpler and simpler calculations. That's what really math is about. So sort of ironic that it's sort of crazy that people find math difficult because it's the language we use to simplify the world, right? And it's a beautiful language at that. So what we're going to do is recreate our circle and we're going to recreate it the same shape that we had as before, but what we're going to do, we're just going to set our numbers to be different, right? And specifically the zero zero points is going to stay exactly where it is. All that's going to change is the distance for our radius we're just going to call that one, right? Because we can, we don't have to make everything too scale, right? So let's do this again. One, two, three, four, five, six, seven, eight, nine, ten, eleven, twelve. Right, so here's our circle and we're going to go nine. One, two, three, four, five, six, seven, eight, nine. And one, two, three, four, five, six, seven, eight, nine. One, two, three, four, five, six, seven, eight, nine. One, two, three, four, five, six, seven, eight, nine. And I'm going to use the same technique since I'm not very good at drawing circles freehand. So I got some floss here and I just put it around this thing and put this point on the center of the circle and off we go. We're just going to draw. I'm just going to keep it tight, right? I'm going to draw our circle and there's our circle. I'm just going to make it a little bit darker so you see it better. So there's our circle and we're going to put the grid on here as well. So I'm going to take my light green. Let's throw our grid on here, our X and Y axis and we're going to call this our X axis. We're going to call this our Y axis and what we're going to do, we're going to draw the circle. We're going to draw the circle with a radius of one, which is going to be the same deal, right? I mean it's going to look identical to what we had before, but I'm just going to call the distance from this origin from the zero-zero point to the edge of the circle as one unit. Nothing stopping me from doing that because it's my drawing, right? Let's see if we've got a pen that was dying off. This one's more solid. Let's do it again. So what we got right now is a circle and as far as we are concerned, we're standing right here, right? At this point and this point has an X coordinate and a Y coordinate, right? So we're going to call this X and we're going to call this Y. So what we're going to do is draw our right angle triangle and we're just going to call the radius of the circle one unit, one anything you want, right? So we're going to call the radius equal to one and I'm going to draw or my triangle. That's my right angle triangle. So the distance here is my X and the distance here is my Y, right? That's how far I've gone along the X, right? So that's the X point right there and this is how far I've gone on the Y, right? What the distance is and that's the length there, right? That's what Y is and keep in mind if our radius is one, this point here is going to be one and zero, right? The X is one and the Y is zero. This point here is going to be zero and one. This point here is going to be negative one and zero because it's a negative X axis, right? Negative X direction and this is going to be zero and negative one and what we have here is this is our terminal arm, right? That's our terminal arm. Terminal side as well is referred to. Our angle theta is this, right? In standard position and usually we don't, when we say angle theta we usually mean standard position. Sometimes they always refer to it as standard position. Sometimes it's just angle. You've gone a certain angle, right? So this is our angle, angle in standard position. This also happens to be our reference angle because that's the angle closest to the X axis, right? Remember the X axis acts like a magnet so it just goes closes that gap. So that's the angle closest to the X axis and our coterminal angle is whatever angle we go around. All we do is whatever this theta is, right? We just add 360 to it or subtract 360 to it multiple times if we want to, right? So this is where we left off, right? With the terminology and the only difference we've done from this circle to the previous circle that we had was we changed the radius, right? And by changing the radius we made our calculation super easy. This is no longer nine or a gazillion or a thousand or whatever you want to think about it. This is one. One is easily easy to deal with, right? Easily scalable. Now what's going to happen is we want to find out as much information about where we are on the circle as we can. So in mathematics what we do, we analyze things. So one of the things we end up doing is looking at the ratios of the sides of the triangle because what's going to happen is, let me grab this guy, what's going to happen is as we move around, as we move around the circle our Y axis, our Y coordinate is going to increase, right? So if we're down here, our Y is zero, right? And as we move up, Y is increasing, right? The Y, part of the coordinate is increasing, increasing, increasing. We get to here and that's one, that's the maximum Y value we can have. And then from there when we start moving down again our Y value is decreasing. It gets to zero here and then from here it goes negative, negative, negative. We get to Y equals negative one and then it goes back up again and where Y goes back up again, zero. So our Y, the range of our Y goes from negative one to one and that's another piece of terminology that we have to learn, which is called the range of a function. Our range refers to what the Y values can be, right? The same thing could be said about the X. If we're standing here, our X value is one and as we move this way along the circle, our X value is changing, our triangle is changing, right? Our triangle is constantly changing, right? So as we move this way, our X value here becomes zero. It's the X part of the coordinate. We keep on going, our X value approaches, reaches negative one. We come back down, our X value goes to zero and then our X value goes to one and so forth and so forth, you know, so far, so far and so forth, right? Or how the phrase goes. So we already use the word range to define what the possible Y values can be. So we can't use the range again for the X value and what we do for that is we call that the domain of the function or the domain of the relation because this isn't a function, right? So what we do is we call the possible X values, the domain and the possible Y values the range, okay? So what we're going to do, we're going to take this down and remember these terms, right? The terminal arm is where you end, right? The side that you're ending, where the angle is being measured to. Angle and standard position is from the positive X axis going counterclockwise. The reference angle is the closest angle to the X axis wherever the thing might be, right? And coterminal is our angle and standard position plus or minus 360 multiple times, right? So the two new terms that we have to learn is the two new terms we learn is the range refers to what are all the possible Y values. The domain is what are all the possible X values, right? Now for a unit circle the range of our unit circle is from negative 1 to 1. Those are all the possible Y values we could have, right? We can't be off the circle, right? We can't go above 1 for Y because that way we're off-planet, right? We're off the circle. We can't go below negative 1, the Y-coordinate, because we're off the circle. Remember the grid is just the grid as a reference point, right? It's something that we refer to. Our relation, our circle, is this. We cannot go off-world. So the range goes from negative 1 all the way up to 1 for a unit circle. So the way we write this is range Y can be greater or equal to negative 1 or less than or equal to 1. As for the domain, the domain of possible X values is the same thing. It goes from negative 1 to 1, right? We can't go off, be greater or X value can't be greater than 1. If it is, we're off-world. We're off the circle and it can't be less than negative 1. So the domain of a function or the domain of the unit circle, this is another function again, the domain of the unit circle, is X is greater or equal to negative 1 and less than or equal to 1, okay? And these come into play in our calculations and it's something that we're going to do a lot of and it'll sink in. Don't worry about it too much. But it is important to realize that we cannot, the X values can't go bigger than 1, less than negative 1, and the Y values can't go more than 1, less than negative 1. So it's numbers between negative 1 and 1, numbers between negative 1 and 1, okay? So how are we going to analyze this? What's one of the first things we're going to look at for a unit circle? And in mathematics, one of the most powerful things in mathematics is ratios. And ratios come into play all over the place. And if you're following the language of mathematics in series four, we're talking about units and ratios and it's the most important thing you're going to have to come out of when you come out of high school mathematics. Just for everyday life, you have to understand ratios and ratios are super powerful and they will use it for the most simple calculations to the most complex calculations. Unit circle is no different. Unit circle allows us to, once we put a right angle triangle on this, allows us to look at what happens to our position on a circle relative to the ratio of the Y axis, the X axis, and the radius. And for a right angle triangle, the radius is called the hypotenuse, right? So we come up with terminology that looks at the different ratios for a right angle triangle, which in turn allows us to understand what happens when we move around the circle. And the three ratios, the three basic essential trigonometric ratios that we have to do, we have to learn is Sokotoa, right, which is the sign of the angle. It relates the angle to the ratio of the sides, the sign of this angle. And again, it's just a terminology that we've come up with, right? I'm sure it's got its origins, some kind of Latin or math mathematicians that came up with this thing. I actually don't know what the origins are. I just know that they're just standardized words that we use to refer to something, right? Like, this is called a pen, right? Or a permanent marker. That's what sign is. Sign is the ratio relating the angle to the sides of a triangle. Okay. So sine theta, sine of an angle is the ratio of the opposite side of that angle to the hypotenuse. And this only works for right angle triangles. So the sign of this angle is going to be Y over R. Y. That's the sign of this angle. It's just the ratio of the opposite side of the angle relative to the hypotenuse. The cos of an angle is X versus the hypotenuse. So cos theta is X versus R. Okay. And 10 of this angle is Y versus X. So 10 theta is Y versus X. Okay. And what these mean is where sine of an angle looks at how the Y, the Y coordinate changes relative to the hypotenuse as you move around the circle. Because if you think about it, if you're here, your Y coordinate, the Y length is going to be small, but the radius is going to stay the same. For this, for a unit circle, it's just one. And as if you get, move around the circle, our Y coordinate is getting bigger, getting bigger, getting bigger. When you get to this point, the Y is one and the radius is one. So sine theta equals one at this point right here. And we're going to explore how the ratios, how to trig ratios change as we move around the circle in a future video. Right now, what's important to understand the grasp is what these terms mean and what they refer to. And if you remember some of the videos, if you've watched some of the language of mathematics, we talked about sine, cosine, tan, right? We refer to sine as opposite over hypotenuse because we weren't talking about a unit circle at that point, right? We didn't throw our triangles on a grid system, right? So there was no Y, X and R. So the Y, the sine of an angle, is also with outside of the unit circle is opposite over hypotenuse, which is what Sokoto is referring to, right? Soka, sine, where are we? Sine is opposite over hypotenuse, cos is adjacent over hypotenuse and tan is opposite over adjacent. So X is adjacent over hypotenuse and tan is opposite over adjacent, right? Now, one of the beautiful things that you have to appreciate about the unit circle is, is this. Since the radius is 1, right? The sine of an angle is Y over R, right? But R for a unit circle is 1. So if we punch in 1 for the radius, this becomes Y over 1, right? Which is pretty trippy because that means for a unit circle, right? Unit circle. For a unit circle, the sine of theta is just equal to Y. That's also true for cos. Cos, if we sub in X, we got X and we sub in 1 for R, this becomes X over 1, which is just your X axis, right? So for a unit circle, Y is equal to sine theta. That's what we get, right? Y is equal to sine theta and for a unit circle, X is equal to cos theta. And this is super important because when you're going around the circle, right? Your coordinate system defines where you are, right? And if Y is sine theta and X is cos theta, then your coordinate system, your X is really cos theta and your Y is really sine theta. So for a circle, for a unit circle specifically, your cos theta and sine theta are your X and Y coordinates. And if that's the case, then you can analyze this thing and realize where you are, depending on the sine of an angle and the sine of an angle and cos of an angle. For example, in this quadrant, let's call this quadrant number one, in this quadrant, our X is positive and our Y is positive. So if cos is positive and sine is positive for a specific angle, you know you're in this quadrant. In the second quadrant, our X is negative and our sine is positive, right? So if our X is negative and our sine is positive, we know we're in the second quadrant, right? Over here, if they're both positive, we know we're in the first quadrant. In the third quadrant, if our cos theta, cos of the angle is negative and our sine, our Y, sine of an angle is negative, then we know we're in this quadrant. And in the fourth quadrant, if our X is positive and our Y is negative, we're in the fourth quadrant, which is absolutely amazing if you think about it, because we started off with a triangle in the circle. And all we did was look at the ratios and all of a sudden we're getting a ton of information coming at us that we can, you know, look at something, look at a cyclic function. And remember, this is all about a cyclic function, right? A circle is the ideal cyclic function. We can start looking at the angle of where we are on the circle and it'll tell us exactly which quadrants we are and what the ratios are of the sides of the triangle, of the sides of the links, the coordinates of the unit circle relative to the radius or the other sides or whatever it is, right? And this is something that it's referred to as what is it called? All students take calculus, right? All means everything's positive here. And remember, the tan is Y over X, right? So the tan is this guy divided by this guy. So if this is positive and that's positive, tan is positive. If we're in this quadrant, the Y is positive and the X is negative, this divided by that makes the tan negative. If we're in here, tan is sin divided by cos, negative divided by negative is positive, tan is sin divided by cos, negative divided by positive is negative, right? So tan is positive, negative, positive, negative, right? All students take calculus, all the trig, the main trig functions are positive here, all students sin is positive here, all students take T, tan is positive here, all students take calculus C, cos is positive here. Now for me, I really didn't use all students take calculus to remember where the sin, cos and tan are positive and negative. All I did was realize that the Y axis is sin and the X axis is cos and sin is positive above the X axis and negative below the X axis and cos is positive to the right of the Y axis and negative to the left of the X axis. So the way I remember where something is positive or negative where the trig identities or trig ratios is not trig identities, the trig ratios are positive and negative is just by realizing that the Y axis is my sin theta and cos axis is my sin theta and X axis is cos theta, right? And that's the way I'm going to refer to it. I don't really use all students take calculus phrase very much. So this is our unit circle. You know, get to know this. We're going to refer to this a lot. We're going to refer to the terminology a lot and the coordinate system a lot and where things are positive and negative a lot. And the other things you have to learn is the range of a function or the range of our unit circle or the range of Y because sin theta is Y, right? So sin theta can only be between negative one and one. Super, super, super important. And the domain which is what are all the possible X values? Which means what are all the possible cos theta values, right? What are all the possible values for cos theta? That can only be between negative one and one, right? So we can write this as cos theta is always going to be greater than negative one or equal to and less than or equal to one. And sin theta is always going to be greater or equal to negative one or less than or equal to one. This is the range of a unit circle. This is the domain of a unit circle, which also means that this is all the possible values that X can, Y can be and negative one to one is all the possible values on what cos theta can be, okay? What we'll do next is we're going to we're going to sort of jump ahead a little bit and I'm dying to do this because it's something brilliant and I love it. And what we're going to do is we're going to graph the sine function relative to how theta changes, okay? So we're going to graph the function sine theta as theta changes, goes around the circle and we're very basically going to see what happens to the Y value as theta as we move around the circle, okay? And it's a beautiful, beautiful wave that I'm sure you've seen and that's the wave that is super important that comes into play in so many aspects of our lives. May it be light coming to us, may it be sound, what you're hearing, may it be vibrations, may it be a myriad of other things. That's what a cyclic function looks like when we graph it relative to it moving around whatever central point it has, okay? I'll see you guys in the next video. Bye for now.