Everyone, welcome back. So let's take a look at this example problem. Let's use what we know about coterminal angles to find the smallest positive angle that's coterminal with the given angles that we see over here, and we're going to sketch them in standard position. Let's take a look at the first one over here, 710 degrees. What does it mean by the smallest positive angle? Well, we could just go on this graph here and go all the way around and then go another rotation and try to figure out where 710 is. But the easiest way to do it is use coterminal angles. If this is bigger than 360, we just have to knock, you know, 360 off. We subtract multiples until we get to something that is between 0 and 360, the smallest positive angle. Alright. So, what is 710? We're just going to subtract 360 because it's bigger than 360. And if you actually subtract this, what you're going to end up getting is you're going to get 350 degrees. So, do we keep going? Do we subtract another multiple? Well, no. Because this is actually less than 360 degrees. So, how do I draw this angle here? This is the smallest positive angle that's coterminal with 710. And this is really just if you just went one full circle around, but actually not quite. You just stopped a little bit short. Right? Because we know that if you go from 0 all the way around the circle, you'll end up back at 360 degrees. So, 350 would be 10 degrees short of a full rotation. So we go going like this all the way around, but you'd stop just around here, and you would end up with an angle that kind of looks like this. So this angle over here is 350 degrees. Alright? And that's how you would sketch that.

Let's take a look at the next one over here, which is negative 37 degrees. Now we know negative angles, you would just draw them clockwise from the x axis, but we actually want to figure out what the smallest positive angle is. Right? That's a negative angle. So what do we do? Well, this is less than 0, so we're just going to add 360 to this. If you add 360 to negative 37, what you'll get is 323 degrees. So, do we keep going? No. Because that's the smallest number between 0 and 360. So this is the smallest positive angle that's coterminal. How do you draw this? Well, again, just use your axis as guides. You have 0, you've got 90, 180, 270, and then back to 360 degrees. So, 323 is going to be somewhere in this quadrant over here. Again, you can use the halfway point to kind of gauge where this is going to be. I draw a line like this, that's going to be halfway between 270 and 360, which is going to be about 315, and that's actually really close to what you want to draw. So you're going to draw a line that looks something that looks like this, but maybe like a little bit nudged to the right. So it's going to look something like that. Alright? So this angle drawn from the positive x axis is 323 degrees.

Let's take a look at the last one over here, which is negative 480. Same thing as example b, we're just going to add 360 to this. Right? So, add 360, and what do you get? You're going to get negative 120. It's still a negative angle. We want the smallest positive, so you just have to add another multiple of 360. So we're just going to add another round of 360 to this. And what you should get is you should get 240 degrees. That is the answer that you want. Alright? So, I'm actually let me go ahead and write this over here. This is going to equal 240 degrees. That's the answer. How do you sketch this? Well, let's just use our guides again. We got 0, 90, 180, 270. So and then back to 360. So 240 is going to be somewhere in this quadrant over here, and 240 is a little bit closer to 270 than it is to 180. Right? The halfway point of this would be 225. So we want something that's a little bit sort of more vertical like this. Alright? So let's go ahead and draw this. This is going to be something that looks about like that. You go all the way around from the positive x axis. That's going to be our angle. That's going to be 240 degrees. Hopefully, this made sense. Hopefully, you get to draw some of these on your own and got something that looks really similar to mine. Thanks for watching, and I'll see you in the next one.