 Check this out. The unit circle is this. Think about this. I've gone into this detail. I want to relate this speedy Gonzales style. Check this out. Here's the thing you have to understand about the unit circle and why we study the unit circle. Us human beings, we're hard wired for problem solving. We want to analyze things. We want to break things down. We want to understand how things work. We want to do that because we want to optimize our lives and try to understand different systems in our world. Initially, mainly associated with survival, because we want to survive, human nature, animal nature, to want to survive. The way you survive is you understand your surroundings better. Mathematics is a language we came up with to try to understand the world around us better. One of the systems that we encounter, which manifests itself in multiple different areas in our lives, is what's called cyclic systems, cycles, things that repeat. That's hardwired in life on this planet because the earth rotates around this axis. The earth rotates around the sun. There are seasons that occur. Our daily lives have routines associated with them. When we looked at the world, mathematical perspective from a mathematical perspective, and when we try to understand the cyclic system, we think to ourselves, well, we need to model this. What's the best way to model a cyclic system? What's something we can come up with that repeats? For us, the thing that we've come up with, the shape that we've come up with that repeats is obviously a circle. We can draw a circle and say, okay, a circle repeats. If you start off here, right, you move in this direction, right? This is you. Depending on how fast you're going, you end up here again, right? And then if you do it again, you end up there again, do it again, end up there again. Wow, this is a cyclic system, an ideal cyclic system because from one understand, there are no such things as perfect circles in nature, right? So we drew, we draw a perfect circle and we say, okay, let this circle represent a cyclic function, a cyclic system, something that repeats itself. You get up in the morning, you brush your teeth, you eat your breakfast, you do whatever it is you're going to do for the rest of the day, you come home, you do whatever it is you do at home for the rest of the day, you go to sleep, you wake up in the morning, you do again to a certain degree, you do again to a certain degree, you do again to a certain degree, you do again to a certain degree, right? Now, let's check it out. I remember doing this simpler math problem set with a situation of being chased by wolves over a mile with an additional wolf every 150 meters. Real question from a Chinese teacher. So if you're doing this and there's a wolf every 150 meters, how often you're going to encounter the wolf, right? That's a cyclic function, right? So for us human beings, when we try to understand a certain type of system, we try to break it down, right? We take things apart, reverse engineer things, right? That's a legitimate thing. You reverse engineer something to be able to recreate that thing, right? The famous story, sometimes when you take things apart, you can't recreate them, right? The goose that laid the golden egg, right? Suppose there was a goose that was laying golden eggs and the owners of this goose were very happy getting a golden egg a day, right? Or a week. They're like, well, we got a chunk of gold, you know, let this duck do its thing and we'll get tons of golden eggs. And then some genius came along and said, hey, let's take that thing apart and make more golden eggs a day. And the owner of the duck went, hey, what a great idea. Let's dissect this thing and find out how he lays golden eggs. They dissected it and didn't know how to make golden eggs and their duck that laid golden eggs was no longer a duck. It was a dissected piece of thing on a table, right? That could no longer lay golden eggs. Well, that wasn't a smart thing to do. Lucky for us, we can take a circle apart. We can break it into pieces and try to understand it better. And the way we break something into pieces, we go, okay, you know what? The most important thing about a circle is it's radius, right? We'll call this R. Okay. And to simplify matters because we human beings like to simplify our problems so we can deal with them better, right? So to simplify this problem, what we do, we say, you know what, let's call R. Let's give R the length of R a number that we can easily scale, right? Now we're not going to give it a length of 94 because that's who wants to deal with a radius of 94 or 72, right? We're going to give R the easiest number that you can scale, 1, right? So we're going to say this is a circle with a radius of 1. This is the beginning stages of a unit circle, length of pi, given the length of pi. The pi appears in this, but we're going to keep the radius simple because if you give this the length of pi, then you're going to have more pies appearing in your circumference, right? Shhh, don't tell people. Froggy, Mando, first time chat, some fun. So we give the length of the circle a unit of 1. More pies, more pies, fun also. Thank you for the follow, Froggy. So we give the radius of 1 a unit of 1. So you know what, this is going to be our standard circle that we're going to scale and try to understand every other circle, and why we try to understand circles because we're trying to understand cyclic functions, right? So what we do is we say, you know what, a circle with a radius of 1, we're going to call the unit circle, unit circle, unit circle, right? Okay, I never actually understood how this pi connected to circles and the significance of it. We'll talk about it, baby, nice. We're going to get to it. Okay, so now that we created a unit circle, we want to understand this unit circle better. So what we do, we break things into pieces. So the ideal thing, the simplest thing to do to break this thing into pieces is, we're going to go through the center of the circle, horizontally, we're going to go through the center of the circle, vertically. So all of a sudden we cut our circle into four different quadrants. And guess what? Now we don't have to understand the full circle. All we have to understand is what happens in this quadrant, and then we can just mirror it and we understand what's happening in the whole circle. We just made this problem a quarter easier, right? But that doesn't sound on the topic, but who knows? That message appears all the time for our channel because it's very important for humanity to free us so much, right? Now we're trying to understand the circle. We do that, right? And since we've broken this up horizontally this way and vertically this way, well we have something that we use that's similar to this. We call this a Cartesian coordinate system, our night bump. We're going to call this the x-axis. We're going to call this the y-axis, right? That's the x-axis and that's the y-axis. And if you want to find out where you are on a circle, right? Well, you can just talk about the coordinates x and y, right? And this takes our circle and links it up to a right angle triangle, which is your distance here is your x and your distance here is your y. Oops, your y, right? That's your x and that's your y, right? And what you end up having is a right angle triangle and our froggy, as Elder God says, Junissons is central to our philosophy here, which is free speech for all and sharing information, right? Fundamental, fundamental, aside from transparency and accountability of capitalist power. And if you remember, so what we've done right now is connected up a triangle to unit circle, right? And if you remember your Pythagorean theorem, there's a formula for a triangle, right? There's a formula for a triangle, for a right angle triangle, which is called Pythagorean theorem, which is a squared plus b squared. Let's make this a small b. It usually appears as a small b, right? b squared is equal to c squared. And for this right angle triangle, you would have x squared plus y squared is equal to 1 squared and 1 squared is just 1. So right now we've got x squared plus y squared equals 1, right? Froggy, I'm a fan, but yeah, no, a lot of your friends in college went the other way, yeah. So x squared plus y squared equals 1. That's one of the formulas we can get from this, right? So if we have a radius of 1 here, and if anywhere we're on the circle, we can derive the coordinates of the step, right? You saw the NSA recruiter in the math department, listen to William Beanie, William Benny, Benny, Benny, the guy who came out with the NSA system, he quit them, he quit them, and he became a whistleblower against them, right? William Beanie, that might be pronouncing his last name incorrectly, right? So this is the fundamental principle of a unit circle, right? Now one thing that happens is for a right angle triangle we have theta here, and theta is in standard position is usually, well it is, the angle that we're moving around the circle relative to the positive x-axis, right? So I could also tell you to go to this point by telling you the angle that you're going to go on the circle, right? Froggy, did you see the British guy online asking a viewer if they even knew the fact? No, I didn't see. Crazy. Oh my god, my company recently introduced the math quiz for all possible new recruits. Oh, the things I have seen, oh man, I can't even imagine, oh my god, I can't even imagine, right? So what we can do is talk about an angle, right? In degrees usually, but another thing we can talk about is eliminating, again math petitions are, what does this say? Pierce Morgan. And then his co-host called her up and he said the question was 3.147 or something, crazy. And Pierce Morgan, well isn't he the guy that doesn't believe in bodily autonomy? He can kiss Matt, right? So what you can do, mathematicians come along, and this is where pi is related, and say you don't want, forget about degrees, right? Forget about degrees. No for, no for any, I have embarrassing math literacy, so they're really cool, much needed a refresher, awesome, awesome. Now take a look. Mathematicians in general, I tell all my students, mathematicians are lazy creatures, right? Because they like to simplify things as much as possible, okay? Simplify to level where they can actually easily solve a problem and then extrapolate all the, everything else that they need to extrapolate. So mathematicians end up looking at this and saying, you know what, we've got too many variables here for the circle. We, you know, we've got an r to find out where you are on here. And theta and degrees, right? You know what, forget about degrees. Mathematicians came along and said, you know what, we're going to take the variable r and theta, we're going to merge them together, and we're just going to really eliminate the theta in degrees. We're going to start coming out with a new unit called theta in radians, and theta in radians is this. If you have a circle, right, and you're going to travel one, oops, that should be a one, let's erase it with the finger, one. If you're going to travel one radian, rad, is that too small? That's too small, let's make it bigger. If you're going to travel one rad, here's this, if you're going to travel one rad radian around the circle, that means that should be more if this is the radius, right? If this is r, one rad, the angle traveling around the circle is the same distance as you travel r around the arc length of the circle, right? So if your radius here was 10 and if you travel 10 meters on the circumference of the circle, we're going to call that one radian, one radian, the angle that you've traveled around the circle, right? So they just eliminated degrees, they just eliminated degrees from the unit circle, right? It's still there, you can use degrees, but they simplify the matter and call it radians, and this is where pi comes into place. No matter the size of a circle, doesn't make a difference what size a circle is, right? If you travel half the circle, the radius along the circumference of a circle, right? If you travel along the circle, half way around the circle, then you've traveled pi radians, you've traveled 3.1415, 3, wow I'm just going to write down pi, pi times r, whatever the radius is here, radius, right? So if I give you a circle, let's say the radius is 10, then how far, what's the distance half way around the circle is 10 times pi, 10 pi. If the radius is 20, then what's half the circle? 20 times pi, no matter what size circle, and that's where pi comes into play, right? It's very cool, this goes a lot deeper as well. I just try to sort of define what a unit circle is and give you a little bit more on it, but we've done a lot of videos on this. If you do Chichou Trigonometry playlist, I have a full playlist on, here let me find it for you, full playlist on Trigonometry, it's basically the first ASMR math videos I started putting together, because it's so important, so important, so crucial. Let's see if it's going to pop up. Trig, there we go. Take a look at this, I'm proud of this playlist. These videos here, you want to understand Trigonometry, these videos will do the trick for you. Really, if you're patient enough to sit through them, these videos will do the trick for you. I'll have the link in the description of the video once it's been uploaded to our platforms. I'm just going to get caught up with the chat here.