Coterminal Angles - Online Tutor, Practice Problems & Exam Prep

To find coterminal angles, add or subtract multiples of 360 degrees. For example, 390 degrees is coterminal with 30 degrees, as both point in the same direction. Similarly, negative angles can be converted to positive by adding 360 degrees, such as negative 270 degrees becoming 90 degrees. For larger angles like 1,000 degrees, subtracting 360 twice yields 280 degrees, which is coterminal with 1,000 degrees. Understanding coterminal angles is essential for graphing and analyzing angular relationships in trigonometry.

Coterminal Angles

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.

Example 1

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.

Here’s what students ask on this topic:

What are coterminal angles and how do you find them?

Coterminal angles are angles that share the same terminal side when drawn in standard position. To find coterminal angles, you add or subtract multiples of 360 degrees. For example, if you have an angle of 45 degrees, adding 360 degrees gives you 405 degrees, which is coterminal with 45 degrees. Similarly, subtracting 360 degrees from 45 degrees gives you -315 degrees, which is also coterminal with 45 degrees. The general formula is: $\theta \pm 360n$, where $n$ is an integer.

How do you find a positive coterminal angle for a negative angle?

To find a positive coterminal angle for a negative angle, you add 360 degrees until the result is positive. For example, if you have an angle of -150 degrees, you add 360 degrees: $-150+360=210$. Therefore, 210 degrees is a positive coterminal angle for -150 degrees. This method ensures that the angle lies within the standard range of 0 to 360 degrees.

What is the coterminal angle of 1000 degrees?

To find the coterminal angle of 1000 degrees, you subtract 360 degrees repeatedly until the result is between 0 and 360 degrees. First, subtract 360 degrees: $1000-360=640$. Then, subtract 360 degrees again: $640-360=280$. Therefore, 280 degrees is the coterminal angle of 1000 degrees.

How do you determine if two angles are coterminal?

Two angles are coterminal if they share the same terminal side when drawn in standard position. To determine if two angles are coterminal, find the difference between them and check if it is a multiple of 360 degrees. For example, to check if 45 degrees and 405 degrees are coterminal, calculate the difference: $405-45=360$. Since 360 degrees is a multiple of 360, the angles are coterminal.

What is the coterminal angle of -270 degrees?

To find the coterminal angle of -270 degrees, you add 360 degrees until the result is between 0 and 360 degrees. Add 360 degrees: $-270+360=90$. Therefore, 90 degrees is the coterminal angle of -270 degrees.