 For your convenience in calculating trigonometric functions, it's often times useful to use a diagram like you see on the screen right now, which is often referred to as the unit circle diagram. This is common for various trigonometry textbooks. You can find it online, multiple sources. The picture I'm using right now in this video is courtesy of the one on the Wikipedia page for the unit circle. What this diagram does for us is it illustrates the special angles that we should have memorized, or at least we have this diagram so we can reference it, so that we can calculate without actually doing calculations. We can just reference the table to see what's going on here. This is our unit circle. We have our first quadrant right here, second quadrant, third quadrant, fourth quadrant, like there on the picture. Since the coordinates, if we take this point right here, how are we supposed to interpret this angle? What this diagram tells you here is that the angle between the positive x-axis and this one right here, this angle theta, which in degree measure is 30 degrees, in radian measure it's pi over 6. We see that the unit circle, that is the circle which has a radius of 1, it'll intersect this line at the point root 3 over 2 comma 1 half. So the x-coordinate is root 3 over 2 and the y-coordinate is 1 half. Well, we have to remember here that the x-coordinate on the unit circle gives us cosine of theta and the y-coordinate on the unit circle gives us sine of theta. So when we look at this arc right here, what we're seeing here is that cosine of 30 degrees or cosine of pi sixth, whatever angle measure you're using here, this is equal to root 3 over 2, the x-coordinate, and sine of 30 degrees or sine of pi over 6, this is equal to 1 half. So that's how one reads this chart right here. So if we picked a different angle, for example, if we pick 45 degrees, this says that cosine of 45 degrees is root 2 over 2 and sine of 45 degrees is root 2 over 2. If we pick the angle measure pi thirds, this says that cosine of pi thirds is 1 half, sine of pi thirds is root 3 over 2. If we take the angle measure 240 degrees or 4 pi thirds, then this tells us that cosine of 4 pi thirds is negative 1 half and sine of 4 pi thirds is negative root 3 over 2. to. So this gives us our special angles and everything that references to those. Now, if you have this printed out, then it feel free to use it, right? But there are some situations like maybe in a test, you don't have this diagram. It turns out we are quite capable of reproducing this diagram from memory. And I don't just mean because we're like some genius or photographic memories, we can actually reproduce it in the following way, right? So the first thing I want to mention here is that when you look over here in the second quadrant, the second quadrant, the x-coordinate is negative and the y-coordinate is positive. So if you take a point like, say, 150 degrees versus 30 degrees its reference angle, the absolute values are the same, root three over two versus one half, through three over two versus one half, right? The only thing that's different is the x-coordinate is negative in the second quadrant. It's the first quadrant, it's of course going to be positive. Both are positive, like so. And so if you know that in the second quadrant, x is negative and y is positive, then you know the exact same values to switch the signs on the x-coordinate from what happens in the first quadrant. In the third quadrant, we do a similar thing. In quadrant three, we get x and y are both negative. So we can take, for example, 210 degrees, like we see right here, the reference angle would be 30 degrees, which over here. In which case then, the numbers in terms of absolute values can be root three over two and one half, just like it is over here. It's just it's a double negative, negative, negative for the coordinates there. Similar in the fourth quadrant, in the fourth quadrant, we see that x is positive, y is negative. So calculate the reference angle and use the sign to compute these. So if you understand how reference angles work and the quadrants work, it turns out you only need to know, you need to only memorize it for the first quadrant. So let's zoom in on that a little bit right here. Okay, so what's the pattern? So you memorize, you go from 0, 30, 45, 60, 90. There's usually not a problem with that. You should also know the corresponding gradient measures, 0, pi 6, pi 4, pi thirds, pi halves. We have that. Now, if you look at the sequence of y-coordinates, that is the sign values, I want you to know it's the following. And I'm going to write these in unsimplified form. If you look at 0, 0 is actually just the number of the square root of 0 over 2. And that might seem silly, but it's true. The square root of 0 is 0, and 0 over 2 is 0. Then if you look at, say, one half, one half is the same thing as the square root of 1 over 1. The square root of 1 is 1, right? And then the other ones, no simplifications necessary, we're going to get the square root of 2 over 2 for 45 degrees. You get the square root of 3 over 2 for 60 degrees. And then you're going to get 1. 1 is the square root of 4 over 2. The square root of 4 is 2. 2 over 2 is 1. And so if you look at this pattern here, it's like, okay, it always looks like a root of something over 2, and it just increases the sequence, 0, 1, 2, 3, 4. Okay. So that's a sequence you probably can easily memorize. Okay, 0, 1, 2, 3, 4. Sign is going up as you go from 0 degrees to 90 degrees. On the other hand, cosine's going to get smaller. Cosine starts at 0 and goes all the way to 1. It's the exact same pattern. It just works backwards, because as you go from the y-axis down, the x-coordinate's going to get larger, or if you're thinking of the other way. And so this mnemonic device can help us kind of memorize this diagram so we can reproduce it on the fly in some type of desert island trigonometry situation, right? But what I want to do for the moment is just show you how you can use this to make a calculation. So let's say we want to do something like, oh, let's do sine of 5 pi over 6. What does that mean? When you look it up on the diagram, 5 pi over 6 is right here. The y-coordinate is 1 half. Great. How do you do cosine of 5 pi over 6? Again, you just look it up here. It's going to be a negative root 3 over 2. How would you do something like tangent? Tangent of 5 pi over 6. Well, tangent, recall, is going to be sine of theta over cosine of theta, where theta here is 5 pi over 6. And so you're going to take the ratio 1 half over negative root 3 over 2. It's a fraction divided by a fraction, so you can multiply it by the reciprocal. So you're going to get 2 over negative square root of 3. And so you end up with negative 1 over the square root of 3. You can rationalize that if you so want to, to be negative 3 over 3. And that would then be how you compute tangent. If you wanted to do something like, well, what's secant? Secant of 5 pi over 6. This is going to be 1 over cosine, which cosine we saw earlier was negative root 3 over 2. So you're going to get negative 2 over the square root of 3, which again, if you need to rationalize it, which is usually a silly thing to do, you're going to get negative 2 over, negative 2 times root 3 over 3, something like that. And so you can compute cotangent and cosecant similarly. So if you have this unit circle diagram, you can compute the six trigonometric functions for all of these special angles. In particular, you want to know the first quadrant, use the pattern I showed you just a moment ago, and then you could reconstruct the rest of it using quadrants and reference angles.