So up until now, we've seen how to graph angles between 0 and 360 degrees. If I go all the way around the circle, that's a full 360 degrees, and 30 degrees is going to look something like this. But what if I were asked to draw an angle that is not between 0 and 360, like this first example, 390 degrees? Well, actually, let's just go and look at the circle and actually do it. Because, really, the idea is that going around the full circle is 360 degrees. So 390 degrees would just be if I continued going around the circle an extra 30 degrees. So 360 degrees plus an extra 30 degrees over here is going to give me 390 degrees. And that actually ends up pointing in the exact same direction as 30 degrees. These are what we call coterminal angles. Now, problems will not always be this simple, so I'm going to show you a really quick way to find coterminal angles with an angle that you're given, especially when you have really big numbers, like 1000 or something like that. Let's go ahead and get started. I'll break it down for you.

Angles are coterminal if they point in the same direction. So that means that they have the same initial or terminal side. Remember, terminal or, sorry, initial means it's going to be along the x-axis. We always have angles that are initial along the x-axis. So that means that they have the same terminal side. Coterminal being the same terminal side. Alright? Now, basically, what we saw here is that the only difference between 30 and 390 degrees is that 390 degrees has gone around a full rotation one time. So, in other words, the difference between them is just a multiple of 360 degrees. And that's the shortcut. To find angles that are coterminal with a given angle, all you have to do is add or subtract multiples of 360 degrees.

So let's take a look at our first example over here, which is 390 degrees. The coterminal angle for that is just going to be 390 degrees minus 360 degrees, and that's just going to be 30 degrees, which is exactly what we saw over here. That's why we got 30 degrees. So let's look at the second example. How would I get something like negative 270 degrees? Well, this angle, sorry, this example specifically says I want an angle between 0 and 360 degrees that's coterminal. So in other words, I want a positive number. So, negative 270 degrees, if I were to graph this, I'd have to go in the clockwise direction until I get all the way out to negative 270 degrees. But what I can just do really quickly here is I can just add 360 degrees over here, and what you're going to get is you're going to get 90 degrees. So in other words, 90 degrees positive angle is going to be coterminal with negative 270 degrees. 90 degrees positive looks like this. Right? It's just sort of straight up like this along the y-axis. So 90 degrees and negative 270 degrees are coterminal. They both point in the same exact direction. These would be in the same direction.

So let's look at the last one over here, which is 1000 degrees. That's a really big number. We wouldn't want to have to go around the circle and around the circle again until we actually ended up at 1000 degrees. So what we can do is really quickly find the coterminal angle by subtracting 360 degrees. If I subtract 360 degrees from here, I'm going to get 640 degrees. Am I done? Do I stop? Well, remember, this is still more than 360 degrees, so I have to subtract another rotation of 360 degrees. So this is actually if I've gone around a full circle more than twice. And so when you do this, what you're going to get is you're going to get 280 degrees. Now that this is a number between 0 and 360 degrees, we can stop here, and we can just only graph what 280 degrees is. How do we do that? Very simple. We've done this before. This is going to be about what 270 degrees looks like, or along the negative x-axis. So, 280 degrees is going to look something like this, just about 10 degrees past that. So this is 280 degrees, or this is also 1000 degrees. They both point in the same direction. Those are coterminal. That's really all there is to it, folks. Thanks for watching, and I'll see you in the next one.