Here, we're going to talk about sketching angles and their reference angles. So, let's start off with an angle. So, if I draw an angle in here. So, when we talk about an angle, we talk about, well, there's a way of measuring the angle; right? So, angle gives you rotation. And if you rotate in the -- let's see, this would be the counterclockwise direction, we say the angle has a positive measurement or a positive measure. So, I have graphed an angle right here of about 40 degrees, okay, positive 40 degrees, where going all the way around, of course, is 360 degrees. Now, of course, if you graph an angle going the other way, right, if you rotate in the clockwise direction, let's say I rotate this far, maybe to here, like, I start here and I rotate like so, that's not in the positive direction. We call that rotation in the negative direction. And I just graphed an angle of about negative 200 degrees; okay? Now, where you start, the side you start from, that's called the initial side. So, the initial side. And the side where you end, that's called the terminal side. Terminal side. And the little point right here where the two sides meet, that has a special name, too, that's called the vertex; okay? So, the vertex in the problem below, right here, the vertex is right there. The initial side is right here, and the terminal side is right there. So, if you rotate in the counterclockwise direction, we see the angle has a positive measurement. And if you rotate in the clockwise direction, the angle has a negative measurement. And in both of these cases, I measured my angles in degrees. Okay. Let's go ahead and find a way of measuring an angle -- or rather sketching an angle in what we call standard position. Okay. So, what is standard position? It's when you make the initial side of your angle -- when you make that side lie along the positive x-axis. Okay. So, this would be the positive x-axis. So, whenever you graph an angle in standard position, you have an x, y coordinate system established. So, let's sketch 60 degrees in standard position. Okay. Now, first of all, I notice it's positive 60, which means I'm going to be rotating in the counterclockwise direction. So, my initial side, my vertex is at the origin. And my initial side corresponds right here with the x-axis. Now 60 degrees; right? It rotate all the way up like so. And 60 degrees might put me maybe right there or so. So, this is a 60-degree angle represented in standard position. Okay. Let's sketch negative 150 degrees in standard position. So, once again, the initial side is along the x-axis, with the vertex at the origin. So, like so. And now it's negative; right? So, we're going to rotate in the clockwise direction? Now, negative 150, that might have me end up right about let's say there or so. So, my rotation is as follows, like that. So, it's negative 150 degrees. Okay. So, you may have noticed that, well, 60 degrees wasn't so bad; right? But negative 150 degrees, that was a little tricky. You might be thinking, well, how am I supposed to know where that is? How do you get an approximation? And notice that if I went sort of halfway around, how far would that be? Well, if I did that, that would be negative 180 degrees; right? So, what I said to myself was, well, I don't want to go all the way, I don't want to go all the way to negative 180. I want to back that off about 30 degrees. And that's how it goes when you're sketching an angle. You want to think in terms of angles you already know that are easy to work with. Like 180, I just took 30 away from 180; right? So, I made this angle right here, this small angle 30 degrees. Now, this small angle has a special name when we're graphing. It's called the reference angle. So, it turns out the reference angle is really useful for graphing an angle. And let's go ahead and define it and see how we use it in further examples. Okay. What is the reference angle? It's the acute angle. Now, remember, acute means between zero and 90 degrees, if you're using degrees, or between zero and pi over two if you're talking about radians. So, basically, it would be between zero and 90 degrees. So, it's the acute angle formed by the terminal side and the x-axis. Okay. Let's see how this works. Sketch the angle theta is equal to 120 degrees, and identify the reference angle. Okay. 120 degrees, well, that's positive; right? So, again, we have the vertex at the origin and we draw in the initial side like so. And then where would the terminal side be? Well, 120 degrees, now that's getting close to 180, right, but not quite. It's 60 degrees away from 180. So, the terminal side is going to be right about there. And so this would be my angle, positive 120 degrees. But notice the way I sketched it. I thought about how far it was from 180. So, what is the reference angle? It's the acute angle formed by the terminal side -- this is the terminal side -- and the x-axis. Okay. So, what's the angle between the terminal side and the x-axis? What's the acute angle? That would be this angle right here. So, that's the angle. It's the reference angle. And how -- and what's that measure? Well, if this is 120, and all the way over is 180, the reference angle in this case must be 60 degrees. So, finding the reference angle is very, very important in trig. So, you want to make sure you understand clearly what it is. It's the acute angle formed by the terminal side and the x-axis. Okay. Let's try another one like that. So, here, we're asked to sketch theta is equal to 5 pi over 6 radians, and identify the reference angle. Okay. First of all, you have to remember that there's a -- you have to remember your conversion between radians and degrees. Pi over 6 radians, right, is 30 degrees. And pi over 4 radians is 45 degrees. And pi over 3 radians is 60 degrees. Okay. These are the basic ones. And, of course, we also have pi radians is 180, and 2 pi radians is 360 degrees; okay? So, these measurements or these conversions should be memorized. So, here, we're asked to sketch the angle 5 pi over 6 radians and identify the reference angle. Now, when you do this, do you remember in the previous example how we compared our angles to, like, 180 and to 360 degrees? We're doing the same thing here, but now I'm not looking for 180 or 360. I'm looking for convenient measures of the -- of radians, which would be, the main ones are these, pi and 2 pi. So, when I look at this, I say to myself, oh, 5 pi over 6, huh, that's pretty close to 6 pi over 6. Why did I think 6 pi over 6? Because I knew 6 divided by 6 was 1; right? This is 1 pi. So, that's pretty close to 1 pi, and 1 pi I know is 180 degrees. So, 5 pi over 6 is really close to 6 pi over 6. So, when I sketch 5 pi over 6, well, I almost go 6 pi over 6, I almost go pi radians, which is 180 degrees, but not quite. I have to back it off how far? Well, what's missing? Clearly 1 pi over 6, but 1 pi over 6 is 30 degrees. So, let's sketch this now. Now it's positive; right? So, here we go. Here's the initial side. And how far do I rotate? I'm going in a positive direction. Well, you want to go 5 pi over 6, which is almost 6 pi over 6, but you're missing 30 degrees. So, I rotate about -- let's see, I stop about where? Maybe about -- let's see, that's a little bit off. Right about there I'd say. So, right there, that rotation is 5 pi over 6. Now, do you see how I drew it? I was thinking about the reference angle right here. This little angle that is between the terminal side and the x-axis, that's the reference angle. So, the reference angle. In that case -- or in this problem, the reference angle's equal to -- you don't want to write 30 degrees; right? It's in radians. So, the reference angle is pi over 6. Okay. So, really important that we're thinking in multiples of pi, 1 pi, 2 pi, 3 pi, negative 1 pi, negative 2 pi, and we understand that whatever's attached to the pi is a fraction, and you're trying to convert your fraction into a convenient number like one or two or three, you know, so on. Okay. Let's try another one like that. Okay. We want to sketch theta, the angle is equal to negative 5 pi over 3, and identify the reference angle. Okay. So, the first thing I think to myself is, huh, negative 5 pi over 3, you know, I want to get that close to, like, either negative pi or negative 2 pi; right? Or, like, negative 3 pi. Now, negative 5 pi over 3, that fraction, right, 5 over 3, that's pretty close to 6 over 3. So, let me write it here. Negative 6 over 3 pi; right? So, it's close. It's off by pi over 3. So, when I sketch, what I'm going to say to myself is, okay, well, first of all, I'm going in the negative direction. So, here's my center. And my initial side is like so. And I'm rotating in the, let's see, it's the clockwise direction; right? Now, I don't want to go a full negative 6 pi over 3; right? I only want to go negative 5 pi over 3. Now, when you compare 5 over 3 to 6 over 3, what's missing is a pi over 3. So, I'm going to go all the way around this way, but not quite. I'm going to not go this far, which is 1 pi over 3. So, here is my terminal side. Now, remember, pi over 3 is 60 degrees; right? So, I drew it intentionally so that this is 60 degrees, or pi over 3. So, here's my angle. It's all the way around in the negative direction. So, my angle is negative 5 pi over 3. And what's my reference angle? Well, it's the angle that is left between my angle, the terminal side of my angle, and the x-axis. So, here, the reference angle -- remember, it's always positive; right? It's acute. So, the reference angle is going to be, well, pi over -- positive pi over 3. Now, you might say, wait a minute, you rotated in the negative direction. That's true, but the reference angle is always positive. It's always a positive acute angle between zero and 90, or zero and pi over 2; in this case pi over 3; okay? So, that's how we sketch angles in degrees and radians in standard position, and the very, very important task of finding the reference angle. This is crucial for us. Okay. So, we'll do more with this coming up soon. Until next time.