In this problem, we're given a completely blank unit circle and asked to fill in all of the missing information. Now this is a specific type of problem that you'll get better at with repetition. It's definitely something that you should try on your own before jumping back in with me. Here, I'm going to walk you through my thought process in filling in the unit circle, and what works best for me might not be what works best for you. So, of course, try this on your own and figure out exactly what method works best for you for filling in the entire unit circle. Let's go ahead and get started here.

Looking at our unit circle, the first thing that I like to do is fill in all of the information for my 4 quadrantal angles because to me it's the simplest information on here. Now looking at this first angle measure, I have 0 degrees or 0 radians and I'm going to go ahead and fill in all of my angle measures for my 3 other quadrantal angles. Up here I have 90 degrees, 180 degrees, 270 degrees, and remember coming back around to a full rotation is 360 degrees. Now oftentimes a blank unit circle will only have one blank there but it's important to remember that this is also a full rotation around. Now let's also fill in those radian angle measures. We have π2 radians, π radians, and 3π2 radians.

We have all those angle measures filled in both degrees and radians, so let's consider our trig values of our quadrantal angles as well. Remember, on the coordinate system, we know that this point is located right here at (1,0) and then we find our tangent value by simply dividing y over x. So, doing that here I have 01 which gives me a value of 0. Then up here for 90 degrees, this is located at the point (0,1) and dividing y over x here gives me 10, an undefined value. Now over here with π radians, we're at (-1,0) and we again get 0 for our tangent. And then down here at 270 degrees we're at (0,-1) which is also an undefined tangent value. We have all of the information for our 4 quadrantal angles.

So now let's focus on all of the other angle measures for all of our other quadrants. Let's start with quadrant 1 here because these are likely the easiest for you to remember as they are for me. Here we're looking at our 3 common angles. We have 30 degrees, 45 degrees, and 60 degrees. I'm also going to go ahead and fill in my radian angle measures here. In this first quadrant, we have π6, π4, and π3.

Now here, let's focus on all of our reference angles. Each of these angles here has a reference angle of 30, so that's going to help me to figure out what these angle measures are. So let's start here in quadrant 2. In quadrant 2, I look at this reference angle in reference to the nearest part of the x-axis, which is at 180 degrees. This is 30 degrees away from that 180, which tells me I can find this angle measure by simply taking 180 minus 30 degrees. This gives me an angle measure of 150 degrees. Now for that radian angle measure, I can do the same thing but just with π. This is π6 radians away, so I take π minus π6 here and that gives me my radian angle measure of 5π6.

Now let's look in our quadrant 3 here and do the same thing. This is in the opposite direction of 180, so here I can take 180 and add my angle to it instead to get that angle measure. This is 30 degrees away because we're still looking at our 30 degree reference angle here. So if I take 180 and I add 30 degrees, that's going to give me an angle measure of 210 degrees. Now I can do the same thing with π, of course. I add π to my angle here which is π6. So π plus π6 gives me an angle measure of 7π6.

Now let's move on to quadrant 4. We look at this in reference to our final angle, our full rotation of 360 degrees, and this is, of course, still a 30 degree reference angle. So here I can take 360 degrees and I can subtract my angle here. This is 30 degrees, so 360 minus 30 gives me 330 degrees. Now, of course, this is also 2π radians, so I can also take 2π and subtract my angle in radians to get the same radian angle measure here. So here, 2π minus π6 will give me 11π6. Now, looking at all of these radian angle measures, you can also choose to memorize these patterns that happen with your radian angle measures.

I want you to go ahead and try that on your own and then check back in with me. Once you have filled in all those angle measures, we can move on to trig values. Remember in quadrant 1, we always start the same way regardless of how we choose to memorize this. We always start with the square root of 2 for every single one of these values. So I can go ahead and fill that in for every trig value in quadrant 1. Once I do that, I use the 123 memory tool, counting 123 in the clockwise direction for those x values and then 1, 2, 3 back in the counterclockwise direction for those y values. Now, from here, we want to fill in our tangent values.

Remember, you can choose to memorize your tangent values, but it's easier for me to just remember that this is y over x. So I can take these values that I already have and simply divide them. Now because they have the same denominator, we're effectively just dividing those numerators. So here in this first one, I divide my numerators of y over x, one over the square root of 3. Now this is the correct value, but remember, it's going to be better if we rationalize our denominator here. So this will give me the square root of 3 over 3 for the tangent of 30 degrees. For 45 degrees, if I take y over x, those numerators, I get a value of 1. And then for 60 degrees, I get the square root of 3 over 1, which is just the square root of 3.

We have all of our trig values in quadrant 1, but we have these 3 other quadrants. Remember, all of our trig values are the same as those of their reference angles, and we only have to worry about the sign. So from here, I'm going to go ahead and fill in all of these trig values, copying them over from quadrant 1 to their respective reference angle in every single quadrant, and then we can worry about the sign after. All of our trig values are copied over to quadrants 2, 3, and 4, but remember the sign of these values may be different depending on what quadrant they're in. So that's where we consider the sign.

Remember the mnemonic device "All students take calculus" telling us the first letter of whatever trig function is positive in that quadrant. So in quadrant 2, we have S. That tells us that only the sine is positive. So only my y values here, root 3 over 2, square root of 2 over 2, and one-half should be positive. So all of my other values are negative, and I can go ahead and fill that in for all of my other values, putting a negative sign on each of them. We can do the same thing in quadrants 3 and 4. Now in quadrant 3, only the tangent is positive and in quadrant 4, only the cosine is positive, so all of my other trig values must be negative.

I'm going to go ahead and copy those negative signs in now based on that and then we'll rejoin together. You should have all of your trig values filled in now with negative signs on the correct values based on our mnemonic device "All students take calculus." So take a chance to double-check all of your values here with mine.

We have completely filled in all of the missing information on our unit circle. Again, this is a problem that gets easier with repetition, so feel free to try this as many times as you need until you have it down. Thanks for watching, and, of course, let me know if you have questions.