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 gonna 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 to figure out where 710 is. But the easiest way to do it is to use coterminal angles. If this is bigger than 360, we just have to subtract 360 degrees. We track multiples until we get to something that is between 0 and 360 degrees, the smallest positive angle. Alright. So what is 710 degrees? We're just gonna subtract 360 because it's bigger than 360 degrees. And if you actually subtract this, what you're gonna end up getting is you're gonna 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 degrees. 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 degrees 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 gonna add 360 degrees to this. If you add 360 degrees 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 degrees. 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 degrees is gonna be somewhere in this quadrant over here. Again, you can use the halfway point to kind of gauge where this is gonna be. I draw a line like this, that's gonna be halfway between 270 and 360, which is gonna be about 315 degrees, and that's actually really close to what you want to draw. So you're gonna draw a line that looks something like this, but maybe like a little bit nudged to the right. So it's gonna 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 degrees. Same thing as example b, we're just gonna add 360 to this. Right? So add 360 degrees, and what do you get? You're gonna get negative 120 degrees. It's still a negative angle. We want the smallest positive, so we just have to add another multiple of 360 degrees. So we're just gonna add another round of 360 degrees 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 going to write this over here. This is gonna equal 240 degrees. That's the answer. How do you sketch this? Well, let's just use our guides again. We have 0, 90, 180, 270, and then back to 360 degrees. So 240 degrees is going to be somewhere in this quadrant over here, and 240 degrees is a little bit closer to 270 than it is to 180 degrees. Right? The halfway point of this would be 225 degrees. So we want something that's a little bit 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.