- Welcome to an example involving coterminal angles. Coterminal angles are two angles in standard position that have the same terminal side. So, if we want to determine if 132 degrees and -588 degrees are coterminal, there are a couple ways of doing this. But let's go ahead and start by sketching these in standard position.

So, to sketch 132 degrees in standard position, the initial side will be along the positive x-axis, and then we'll rotate counterclockwise 132 degrees. So, from here to here would be 90 degrees, so we'll have to rotate another 42 degrees counterclockwise. So, might be somewhere in here. And, again, just to keep track of this, this was 90 degrees here, and this was 42 degrees here.

Now we'll go ahead and sketch -588 degrees. So the terminal side will be along the positive x-axis. But now we're going to rotate clockwise 588 degrees. We know that one complete revolution is 360 degrees, so let's go ahead and rotate clockwise 360 degrees to here. So we have -588 degrees is equal to -360 degrees plus another -228 degrees.

So we'll have to rotate another 228 degrees clockwise. So, from here to here would be another 180 degrees, and now we'll have to rotate another 42 degrees clockwise. And that actually brings us to the same terminal side as 132 degrees. We know that because we rotated another 48 degrees here, and 48 degrees + 42 degrees does give us the 90 degree angle that we would have here formed by the +y-axis and the -x-axis.

So, yes, these two angles are coterminal. So the idea here is if two angles are coterminal, if you start with one of them, you would have to add and subtract multiples of 360 degrees to obtain the other coterminal angles. What I mean by that is, if we start with one of these angles—let's say we start with -588 degrees.

If we add multiples of 360 degrees and end up with an angle that measures 132 degrees, they would be coterminal. So, if we add 360 degrees to this, that would give us -228 degrees, which is coterminal to -588. But we want to see if we can come up with 132 degrees. So, if we add 360 degrees again, notice how we do obtain 132 degrees.

If we didn't come up with 132 degrees, then the angles would not be coterminal. If we do, then they are. So, more generally, if we want to determine a coterminal angle with any angle theta, the coterminal angle would have to measure theta + 360 degrees × k, where k is some integer. So, if k was negative, it would be the same as subtracting multiples of 360. And if k was positive, it would be the same as adding multiples of 360 degrees.