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; thus, -270 degrees becomes 90 degrees. For larger angles like 1000 degrees, subtracting 360 twice yields 280 degrees, which is coterminal with 1000 degrees. Understanding coterminal angles is essential for graphing and analyzing angles effectively.

Coterminal Angles

So up until now, we've seen how to graph angles between 0 and 360. If I go all the way around the circle, that's a full 360, 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, just let's 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 would just be if I continued going around the circle an extra 30 degrees. So 360 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. These are what we call coterminal angles. 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 is 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 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 coterminal with a given angle, all you have to do is add or subtract multiples of 360. Alright?

So let's take a look at our first example over here, which is 390. The coterminal angle for that is just going to be 30, 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? Well, this example specifically says I want an angle between 0 and 360 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. But what I can just do really quickly here is I can just add 360. Actually, I'm going to go ahead and do this in purple. I'm going to add 360 degrees over here, and what you're going to get is you're going to get 90 degrees. So in other words, a 90 degrees positive angle is going to be coterminal with negative 270. 90 degrees positive looks like this. Right? It's just sort of straight up along the y-axis. So 90 and negative 270 are coterminal. They both point in the same exact direction.

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 what we can do is really quickly find the coterminal angle by subtracting 360. If I minus 360 from here, I'm going to get 640 degrees. Am I done? Do I stop? Well, remember, this is still more than 360, so I have to subtract another rotation of 360. So this is actually going to be 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, we can stop here, and we can just graph what 280 degrees is. Alright? How do we do that? Very simple. We've done this before. This is going to be about what 270 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 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.

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 key is that coterminal angles point in the same direction on the unit circle.

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 degrees. Therefore, 210 degrees is a positive coterminal angle for -150 degrees. This method ensures that the angle points in the same direction on the unit circle.

How do you determine coterminal angles for large angles like 1000 degrees?

To determine coterminal angles for large angles like 1000 degrees, you subtract 360 degrees repeatedly until the result is between 0 and 360 degrees. For example, 1000 - 360 = 640 degrees, and 640 - 360 = 280 degrees. Therefore, 280 degrees is coterminal with 1000 degrees. This process simplifies the angle to a more manageable value while maintaining the same direction on the unit circle.

Why are coterminal angles important in trigonometry?

Coterminal angles are important in trigonometry because they help simplify the analysis and graphing of angles. By converting angles to their coterminal equivalents within a standard range (0 to 360 degrees), it becomes easier to apply trigonometric functions and understand their properties. This simplification is particularly useful in solving problems and understanding periodic functions, as coterminal angles share the same sine, cosine, and tangent values.

Can you provide an example of finding coterminal angles for an angle like 390 degrees?

Sure! To find coterminal angles for 390 degrees, you can subtract 360 degrees: 390 - 360 = 30 degrees. Therefore, 30 degrees is coterminal with 390 degrees. Both angles point in the same direction on the unit circle. You can also find another coterminal angle by adding 360 degrees to 390: 390 + 360 = 750 degrees. Thus, 750 degrees is also coterminal with 390 degrees.