A common problem you may be faced with is filling in an entirely blank unit circle, and being able to do that starts with the first quadrant. So here we're going to fill in all of the missing information in the first quadrant of this unit circle. The way that I do this might not be the way that you choose to do this, and that's totally okay. Always find what works best for you. And here, I'm going to walk you through my thought process of filling in this first quadrant. Feel free to pause here and try this on your own before jumping back in with me. The way that I like to start with this unit circle is to start with my x and my y axes because this is the easiest information for me to remember. So starting with 0 degrees, I know that this point is right here on my x axis at 1.0. Then going up to this other axis my y axis this is at 90 degrees which I also know is π2 radians and is located at the point on the y axis 1.0. From here, now that I have my information on each axis, I like to move on to my degree angle measures because this is also something that's rather easy for me to remember. I know that I start with 30 degrees then 45 degrees and then 60 degrees.
Now here is where it usually gets a bit trickier because I'm going to move on to our radian angle measures, which can be a bit more difficult to remember because you're probably more familiar with degrees than radians. You could choose to use a formula to convert these degrees into radians, or you can kind of reason it out using the information you have on your unit circle, which is what I'm going to do here. So here I'm going to start with 45 degrees. I know that 45 degrees is halfway between 0 and 90 degrees, so that means that that radian angle measure needs to be halfway between 0 and π2. Half of π2, if I multiply those together, that gives me a value of π4. The radian angle measure for 45 degrees is π4. Then moving on to 30 degrees, 30 degrees I know is one third of 90 degrees, so I know that that radian angle measure has to be one third of π2. Multiplying that out, that gives me a value of π6, which is what my radian angle measure is here, π6. Now moving on to 60 degrees, I know that 60 degrees is 2 times my 30 degree angle measure. So that means my radian angle measure is going to be 2 times π6. Multiplying π6 times 2 gives me π3. The radian angle measure for 60 degrees is π3.
You can also choose to just memorize what these radian values are if that's more your style. So here, we would want to think about what are the denominators because all of the numerators are the same; they're all π. So you can kind of think about it as counting down: 6, 4, 3, 2 for your denominators, of course skipping 5 there. It can be a little bit tricky to memorize, but remember, you can always reason it out as we did here. Now that we have all of those angle measures filled in, we can move on to really the meat of the unit circle: the cosine, sine, and tangent values, all of our trig values, or in this case, also our x and our y values. So let's start there with our x and y values or our cosine and sine values.
Now to do this, remember we're always going to start the same way with the sqrt of something over 2, not sqrt of 2, sqrt of something over 2. And I'm going to do this using the 123 method because that's easiest for me to remember, but feel free to use the left-hand method or the left-hand rule or whatever else works. So, remember with the 123 rule, we're going to start with this x value up here and count 1, 2, 3 clockwise, and then go back 1, 2, 3 counterclockwise, and those are our cosine and sine values. Remember we can simplify a bit here because the sqrt of 1 is just 1. So for both of these instances of sqrt of 1, that just becomes 1. Now that we have our cosine and sine values, we can go ahead and find our tangent.
Now remember the tangent of any angle, we can just use our information that we already know because the tangent of an angle is the sine of that angle divided by the cosine of that angle. Now, knowing what we know about our x and our y values, this is also just equal to y over x, which we can find by looking at our unit circle right here. Now looking at this first value, 30 degrees, identifying the tangent of 30 degrees, I have one half and sqrt of 3 over 2 as my sine and my cosine value. So I want to take my sine one half and divide it by my cosine sqrt of 3 over 2. Now because these have the same exact denominator, I'm effectively just dividing those numerators. So really, this just gi