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. 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. Let's also fill in those radian angle measures. We have π2 radians, π radians, and 3π2 radians. Now we have all those angle measures filled in 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 0 over 1 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 1 over 0, 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. Okay, 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 go ahead and focus on all of our reference angles. All of these angles here have a reference angle of 30, so that's going to really help me to figure out what these angle measures are.

Let's start here in quadrant 2. Now in quadrant 2, I look at this reference angle in reference to the nearest part of the x-axis, which is at 180 degrees. So 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. For that radian angle measure, I can actually do the same exact thing but just with pi. This is π6 radians away, so I take π minus π6 here and that gives me my radian angle measure of 5π6. This will work for any angle in this quadrant because we can just do that for any angle. We just subtract our angle from 180 and from pi.

Now let's look in our quadrant 3 here and do the same exact thing. This is in the opposite direction of 180, so here I can actually 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. I can do the same thing with π of course. I add π to my angle here which is π6. So π + π6 gives me an angle measure of 7π6.

Now let's move on to quadrant 4. Here 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. So 360 minus theta, my angle. Here this is 30 degrees, so 360 minus 30 gives me 330 degrees. This is also 2π radians, so I can also take 2π and subtract my angle in radians to get the same to get my radian angle measure here. So here, 2π minus π6 will give me 11π6x.