Coterminal Angles - Video Tutorials & Practice Problems

Coterminal Angles

So up until now, we've seen how to graph angles between zero and 360. If I go all the way around the circle, that's a full 3 60 30 degrees is gonna look something like this. But what if I were asked to draw an angle that is not between zero and 360 like this first example, 390 degrees. Well, actually, it's just look, let's just go and look at the circle and actually do it because really the idea is that going around a full circle is 360 degrees. So 390 would just be if I continued going around the circle, an extra 30 degrees. So 3300 and 60 plus an extra 30 degrees over here is gonna give me 390 degrees and that actually ends up pointing in the exact same direction as 30. These are what we call co terminal angles. Now, problems won't always be this simple. So I'm gonna show you a really quick way to find co terminal 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. So angles are co terminal 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 is gonna 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, co terminal means same terminal side. All right. Now, basically what we saw here is that the only difference between 3390 is that 390 has gone around a full rotation one time. So in other words, their difference between them is just a multiple of 360. And that's the shortcut to find angles that are co terminal with a given angle. All you have to add or subtract multiples of 360. All right. So let's take a look at our first example over here which is 390. The code terminal angle for that is just gonna be negative 300 minus 360. Uh And that's just gonna be 30 degrees, which is exactly what we saw over here. That's why we got 30 degrees. So let's go to the second example, how would I get something like negative 270? Well, this angle, sorry, this example specifically says I want an angle between zero and 360 that's co terminal. 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. But what I can just do really quickly here is I can just add 360. Actually, I'm gonna go ahead and this in purple, I'm gonna add 360 degrees over here. And what you're gonna get is you're gonna get 90 degrees. So in other words, 90 degrees, positive angle is gonna be co terminal with negative 270 90 degrees. Positive looks like this, right? So stay sort of straight up like this along the y axis. So 90 negative 270 are co terminal, they both point in the same exact direction. All right, these would be 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. So we can do is really quickly find the co terminal angle by subtracting 360. If I minus 360 from here, I'm gonna get 640 degrees. Am I done? Do I stop? Remember this is still more than 360. So I have to add another rotation or I have to subtract another rotation of 360. So this actually be if I've gone around a full circle more than twice. And so when you do this, what you're gonna get is you're gonna 280 degrees. Now that this is a number between zero and 360 we can stop here and we could just only graph what 280 degrees is right? How do we do that? Very simple? We've done this before. This is gonna be about what 270 looks like along the negative X axis. So 280 degrees is gonna look something that looks 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 co terminal. That's really all there is to it folks. Thanks for watching and I'll see you in the next one.

Example 1

Everyone. Welcome back. So let's take a look at this example problem. Let's use what we know about co terminal angles to find the smallest positive angle that's co terminal 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 co terminal angles. If this is bigger than 360 we just have to knock, you know, 360 off. We just track multiples until we get to something that is between zero and 360 smallest positive angle. All right. So what is 710? And we're just gonna subtract 360 because it's bigger than 360. 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 co terminal 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 zero all the way around a circle, you'll end up back at 360 degrees. So 350 would be 10 degrees short of a full rotation. So be 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. All right. And that's how you would sketch that. Let's 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 uh 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 zero. So we're just gonna add 360 to this. You add 360 to negative 37 which you'll get is 323 degrees. So do we keep going. No, because that's the smallest number between zero and 360. So this is the smallest positive angle. That's co terminal. How do you draw this? Well, again, just use your axis as guides. You have zero, you've got 91 82 70 then back to 360 degrees. So 323 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 2 73 60 which is gonna be about 315. And that's actually really close to what you wanna draw. So you're gonna draw a line that looks something that looks like this, but maybe like a little bit nudged to the right. So it's gonna look something like that. All right. So this angle drawn, 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 gonna add 360 to this, right? So add 360 what do you get? You're gonna get negative 100 and 20. 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 gonna 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. All right. So I'm actually let me, let me go ahead and write this over here. This is gonna equal 240 degrees. That's the answer. How do you sketch this? Well, this is use our guides again. We got 0 91 82 70. So, and then back to 360. So 240 is gonna 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 to this would be 225. So we want something that's a little bit sort of more vertical like this. All right. So let's go ahead and draw this. This is gonna be something that looks about like that you go all the way around from positive X axis. That's gonna be our angle. That's gonna be 240 degrees. Hopefully, this ain't spent since, hopefully 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 the next one.