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. Now 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 am going to walk you through my thought process of filling in this first quadrant. Now feel free to pause here and try this on your own before jumping back in with me.
Now 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. 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, which is at 90 degrees π12 radians and is located at the point on the y-axis 1. So 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 already 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 π12. And half of π12, if I multiply those together, that gives me a value of π14. So that radian angle measure for 45 degrees is π14.
Then moving on to 30 degrees. I know 30 degrees is 1 third of 90 degrees so I know that that radian angle measure has to be 1 third of π12. Now multiplying that out, that gives me a value of π16, which is what my radian angle measure is here, π16.
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 π16. So multiplying π16 times 2 gives me 2 π16 or simplifying that fraction, π13. So that radian angle measure for 60 degrees is π13.
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 exact way with the square root of something over 2, not the square root of 2, the square root of something over 2. And I'm going to do this using the 123 method because that's the easiest for me to remember, but feel free to use the left-hand method or the left-hand rule or whatever else works. So let's go ahead and get started here.
Now 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 square root of 1 is just 1. So for both of these instances of square root 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 the square root 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 the square root of 3 over 2. Now because these have the same exact denominator, I'm effectively just dividing those numerators. So really, this just gives me a value of 13. Now rationalizing this denominator, this gives me a value of 33 as that tangent.
Then moving on to 45 degrees. I have the square root of 2 over 2 and the square root of 2 over 2, the same exact values for sine and for cosine. So when I divide them, I'm simply going to get a value of 1 as that tangent. Then looking up here at 60 degrees, I have the square root of 3 over 2 and square root of 1 over 2. And remember that when they have the same denominator, we're effectively just dividing those numerators. So I can take the square root of 3 divided by 1 to get me my tangent. So the square root of 3 over 1 which is really just equal to the square root of 3.
Now we don't want to forget those values on our axes here. Remember, we still need to identify these tangents. So here down at 0 degrees, I would take a 0 divided by 1, which is just a 0. So the tangent of 0 degrees is 0. And then up here at 90 degrees, I would take 1 divided by 0 which is not a number at all. And that leaves me with an undefined value up here. And now we have completely filled in all of that information in the first quadrant.
Now when dealing with something like the unit circle, repetition can be really helpful. So feel free to try this problem over and over again and check in with me as needed. Thanks for watching, and, of course, let me know if you have any questions.