Hey, everyone. We now know all of our trig values of our 3 common angles in quadrant 1. But the unit circle has 3 other quadrants, all of which we'll need to know trig values in. Now this can seem really overwhelming at first because there are a ton more angles here that we haven't even looked at yet. But you don't have to worry because for all of these angles not in quadrant 1, we can simply use what's called a reference angle. Every single one of these angles corresponds to an angle in quadrant 1 that we already know absolutely everything about, allowing us to find our trig values really quickly and easily for all of these other angles. But before we even think about trig values, we first need to know how to even find a reference angle. So here, I'm going to show you how to do just that. Let's go ahead and jump right in here.

Now like I said, for angles that are not in quadrant 1, they all correspond back to angles that are in quadrant 1. So every single one of these angles here has a reference angle of 30, 45, or 60 degrees. Now how exactly do we find that reference angle? Well, let's take a look here at our 150 degree angle. We know that this angle measures 150 degrees away from our zero degree measure there. But what if instead we were to measure this angle from the other side of the x-axis, here at 180 degrees? Well, this angle formed with this side of the x-axis from 150 degrees to 180 degrees is only 30 degrees. And if I were to draw this triangle out, we actually see that our 150 degree angle has a reference angle of 30 degrees. And this is how it's going to work for all of these angles. We're always going to measure them to the nearest part of the x-axis and write this angle measure as a positive number. So this angle formed here is a positive 30 degrees and that represents its reference angle.

Now let's take a look at our remaining angles in this quadrant 2, starting with 135 degrees. Now we want to measure this again to the nearest part of the x-axis, which still happens to be this 180 degrees over here. Now from 180 to 135 degrees, this measures 45 degrees away so it has a reference angle of 45 degrees, which again we can see that it forms the same exact triangle just flipped in the opposite direction. Now our final angle in quadrant 2 is this 120 degrees, which, when I take a look at this angle and measure it to the nearest part of the x-axis, again still 180 degrees, this is 60 degrees away from 120 down to 180. So this tells me that 120 degrees has a reference angle of 60 degrees. And again, we can take a look at that triangle and verify that those are the exact same triangle just flipped in the other direction.

Now we want to take a look at our remaining 2 quadrants, quadrant 3 and quadrant 4. And we're going to do the exact same thing here, measuring to the nearest part of the x-axis to determine that reference angle. Now remember, we're going to write these as a positive number because it doesn't matter what quadrant it is. This is always going to have a positive reference angle. So let's first take a look at 210 degrees and 330 degrees. Now for 210 degrees here, looking at its nearest x-axis, here the nearest x-axis is still that 180 degrees. And I see that it does measure 30 degrees away from that. So it has a reference angle of 30 degrees. Now looking at 330 degrees, its closest x-axis is coming around to a full rotation at 360 degrees which is also 30 degrees away. So it too has a reference angle of 30 degrees.

Now let's take a look at 2 more angles here. We have 225 degrees and 315 degrees. Now again, measuring to their nearest point on the x-axis, looking at 225 and 315, these are directly in the center angle-wise in those quadrants the same way that 45 degrees is. So when we look at that angle measure, we see that those are both 45 degrees away from their nearest x-axis. So they both have a reference angle of 45 degrees. Now for our two remaining angles here, we have 240 degrees and 300 degrees. Now again, measuring to the nearest part of the x-axis as we have every other time, we see that these are both 60 degrees away from their nearest x-axis respectively, so they each have a reference angle of 60 degrees. Now you might see that I've color-coded these here, and this is also a way that's going to allow us to kind of remember what all of our reference angles are. So whenever I'm remembering reference angles, I like to think of them as forming an x. When I look at all of my 30-degree angles or my angles that have a reference angle of 30 degrees, they form a perfect x with that reference angle 30. Now the same thing is true of our 45 degree angles. Looking here, all of these angles that have that reference angle of 45 form a perfect x. Now the same thing is true of all of my angles that have a reference angle of 60 forming a taller, thinner x.

Now the other way that you can think about this is by looking at these radian angle measures because all of the ones that have a corresponding reference angle have the exact same denominator. So all of our 30 degree reference angles have a denominator of 6 when we look at this here. Then for 45 degrees, they all have a denominator of 4 when looking at that radian angle measure. And finally, forKey all of my 60 degree reference angles which have a denominator of 3. Now that's a couple of different ways that you can choose to think about that, and now we know what we need to know about finding reference angles. So thanks for watching, and I'll see you in the next one.