 Hi, this is Chichu. Now what we're going to do in these set of videos is talk about trigonometry, and if you're interested in just doing simple trigonometric calculations and knowing what the basics of geometry is, and in 2007 I put a bunch of videos together for the language of mathematics in series one that explore some of the basic concepts of trigonometry and geometry and triangles and similar and congruent triangles and whatnot, right? Some of the basic concepts of trig ratios, but those videos sort of came at you sort of out of the blue. There wasn't any prep work in the introduction of why it is that we do study triangles. Okay, and that's what we're gonna do right now. We're gonna pick up the topic of trigonometry and we're going to start from the beginning and take it all the way up to basically talking about trig ratios, trig functions, graphing trig functions, and talking about trig identities a little bit. Okay. Now as for why we study triangles, the mathematics of triangles basically came about, I think it was I got third century or something like this. It came about as we talked about before in the first first set of videos. Basically us exploring the world around us, right? Astronomy had a huge part to play in this, but trigonometry is basically the concept of studying polynomials that have three angles, three sides, right, or shapes that have three sides and Triangles come into play in our everyday lives and a lot of different places, right? Architectural work, it comes into play in surveys. It comes into play in tripods, basically three. Tripod is the most stable sort of a structure that we can create when it comes to having one focal point, right? That's why most cameras, most video equipment, they're sitting on tripods or drum symbols, right? Drum equipment. So triangles come into play in our lives in a lot of different places. One of the places that I've used triangles, right angles triangles specifically, is to do some survey work where you know I'd be going out doing geophysics and I would have to lay out a grid and what I would do is use special right angle triangles specifically where you know the multiples are easy to calculate, right? Three, four, five and multiples of that, right? So you know we'd lay out a triangle or I would do it solo with you know two or three different tapes and stakes into the ground and basically with line of sight I would set up a grid which was extremely accurate. So that's one area that triangles come into play which is just every day type of thing and sort of different types of calculations, okay? But one of the main important areas where triangles come into play which is never really introduced in beginning level introduction to trigonometry when we start talking about triangles is because triangles are related to circles which is a weird concept to comprehend, right? And the reason why we study circles is because circles are the perfect cyclic function and cyclic functions for us are huge, right? It may be from the life cycle, may it be from the earth rotating around its axis, the earth rotating around the Sun, the Sun going around the galaxy, may it be the ties, may it be the female menstrual cycle, may it be the cycle of work, may it be it may be married of other things. So many things are dependent on a cyclic function and that is for me anyway, the main reason that we study triangles because triangles give us an understanding of how cyclic functions work and cyclic functions aside from every day type of life, they're actually embedded within nature, within light, within sound, within vibrations, right? All of that stuff is basically trigonometry and trig functions where we're graphing waves. That's why we really study triangles and write angles, triangles specifically. So what we're going to do right now is I have some grid paper here and some blank paper on this side. So what we're going to do is draw a circle on this grid paper and we're going to see how triangles are related to circles and why they allow us to analyze cyclic functions and help us to move around the cyclic function, okay? As far as drawing things go, I have a few different colors of pens and stuff. So what we're going to do is we're going to try to go with a brighter color for our triangles, sort of go lighter with the grid, our X and Y coordinate, because it's not enough for us to draw a circle. We need to do measurements. We need to be accurate with our calculations, right? And the way we become accurate with our calculations is we lay down a grid on top of any shapes, any geometric shapes that we work with and that's how coordinate geometry really came about, right? We would study shapes. For example, if you're driving a car, your car usually has four wheels, right? For the wheel to drive straight, you need to have all those tires exactly the same shape or wagon wheels or whatever it might be. And the only way for us to really create things that are identical is to put them on a grid system, is to put them on a coordinate system and do exact measurements. So what we're going to do is draw a grid and throw our circle on there and see how right angle triangles are related to circles, which are basically our ideal cyclic functions. Let's see what color should we use. Let's see how these things come out. Let's see if we can use a light green for this. Let's see how the light green turns. We'll use a blue. So the most important thing about a circle is its center, right? If we have the center of a circle and if we have this radius, that's basically a circle, right? We can recreate that circle anywhere we want, right? So what we're going to do is put a dot almost in the middle of this. Let's try to make it as center as possible. So what have we got? We've got one, two, four, five, six, seven, eight, nine, 10, 11, 12, 13, 14, 15, 16, 17, 22, 23, 24. So we want 12. So one, two, three, four, five, six, seven, eight, nine, 10, 11, 12. So we're going to throw our center of the circle right here. And let's make it center according to the camera. So we're going to throw it right here. OK, that's the center of our circle. And what we're going to do is we're going to go out. Let's see, one, two, three, four, five, six, seven, eight, nine. Let's go out nine squares on the axes. And then we'll sort of rough sketch draw in our circle, right? So one, two, three, four, five, six, seven, eight, nine, one, two, three, four, five, six, seven, eight, nine. And one, two, three, four, five, six, seven, eight, nine. Trick is going to be to be able to draw a perfect circle as close as we can get. And there's there is a joke with the math community that the only people that actually know how to draw a perfect circle are people who are insane, freehand anyway. And I've been able to pull it off, you know, just a handful of time. So I guess those were the moments that I was technically insane or mathematically insane. I'm not sure if that's the case right now, but this is the way we're going to do it. So that's not the best circle, but it's something we can work with, right? Sort of a warp circle. Should we try it again? Let's try it again. Let's see if we can do better, right? So center one, two, three, four, five, six, seven, eight, nine, ten, eleven, twelve. And we're going to go one, two, three, four, five, six, seven, eight, nine. Now the problem with this is if you go diagonally it's nine squares, right? right? So we're gonna have to try to do this free hand. Okay, that is pathetic. So what I'm gonna do, I'm gonna bring a string. So what I'm gonna do is to try to draw this as accurately as possible. I want to grab some floss and we're just gonna use floss as our string to draw accurately. So let's try this again. Yeah? 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12. 1, 2, 3, 4, 5, 6, 7, 8, 9. 1, 2, 3, 4, 5, 6, 7, 8, 9. 2, 3, 4, 5, 6, 7, 8, 9. 1, 2, 3, 4, 5, 6, 7, 8, 9. There's our circle or end points and what we're going to do is tie this up. So I'm just putting a knot on it and to suck her up. We've got our accurate point and what we're going to do is just hold this at the center and just go around, right? Pull it as accurate enough. That's not bad. So we're just gonna go over it and we'll keep our floss for circles that we're gonna create later, right? So this is our circle, this is the center, our origin and we have a radius of five squares or nine squares here, right? So what we're gonna do as well is let's throw a grid on here. Actually put a grid of black marker. Okay, let's see if we can, because well, no, no good. So let's put a grid of light green, coordinates of a light green on here and what we're gonna do is put an x-axis. Hopefully that comes out okay. Let's see if we can be accurate. We'll use a level. That's our y-axis and this is gonna be our x-axis. So we've got a coordinate system on here right now. That's our y-axis and that's gonna be our x-axis and a circle represents a perfect cyclic function and what a cyclic function is, it's something that repeats itself over a certain period of time or over certain distance. So something that constantly repeats, right? It does a full period and it comes and again and does it again and does it again and does it again, right? You can think of a clock, 12 hours, right? Goes into 12 hours and it kicks into p.m. I guess and it goes around again and it keeps going around and around and around, right? May it be the tides of the ocean based on the moon cycle, right? May it be the earth rotating around this axis. May it be sound waves coming at you because sound and light and vibrations, these are all cyclic functions. They're basically waves, right? Light travels as a wave. Sound definitely travels as a wave, right? The only reason you can hear me right now is because speakers are vibrating at a certain frequency and you're hearing me, right? And those are waves coming at you and waves are cyclic functions and I find I'm able to manipulate the period, the amplitude and I can lay one wave on top of another. Those are different types of sounds I can create, right? And the most basic must rudimentary cyclic function is a circle and what we do with circles, what we what we have to do with the circle is we have to understand how to move around it, right? We have to do an analysis of how do we move around this? We need to find exact points on a circle, right? So we'll call this our x-axis and we'll call this our y-axis and this is just a naming system that everybody goes with, something standardized that we all agreed upon, right? We agreed that the horizontal would be the x-axis and the vertical would be the y-axis. So let's do the rest of our drawings with a red and a blue color, okay? Now just imagine if we were on the circle and what you have to keep in mind is this is something that a lot of people make a mistake with, right? The grid is not our circle. The grid is a reference point, right? So when we say we want you to move around a circle or find yourself on a function, you can't go off the relation, right? You can't go off a function if you have a function. So what we do is let's assume we're standing right here, right? We're at the edge right at the corner here of the circle and hopefully that little guy comes out, right? It's supposed to be a little man standing here. So let's assume this is us, right? And we want to move around the circle. Now there are two ways, two main ways that we can find ourselves around the circle at any given location, right? Accurately. What I could tell you is I could tell you that I want you to move along the circle a certain distance at a certain angle from the center, right? That's one way I can tell you to go anywhere on the circle that I want you to go, right? So let's assume, you know, I wanted you to travel up to here, right? Up to this point. So now I want you to stand here. So you're going to go along until you get to this location. Now how am I going to get you to go to that location? The way I'm going to tell you to do this is is to move along the circle at a certain distance, at a certain angle, right? So let's throw a line here from the center to where you are right now. So what we have right now is you've traveled nine units, whatever the units might be. Each one of these squares is, you know, I haven't measured it. I don't know what each one of these squares is on the grid paper. It could be one kilometer. It could be one meter. It could be one mile. It could be a yard, whatever it is, right? So right now we know from the center of the circle to this point, this is called the radius, and this is equal to nine, right? Because we set up our circle with a radius of nine. And based on our grid, our coordinate system, we're going to call this location zero, zero, right? Just as a reference point, it's the easiest way to do this, right? We're going to call the center of the circle zero, zero. So that's one way of telling you to go here, but that becomes a little difficult because measuring the arc length of the circle is difficult. It's not an easy task. It's not just a straight line. You would have to give, you know, the thing a curve. Where are we? A curve from here going all the way around. So if you were able to do that, I would tell you to go a certain angle theta here, right? And theta is just, I believe, is Greek alphabet that we use to represent the angles. You could call that anything you want. It doesn't have to be theta, but theta is a standard symbol that we use to represent the angles, right? Now, that's one way for you to get to here. Another way that I can get you to go to this location is by giving you, since we have an x, y coordinate system set up here, is by giving you a coordinate system, right? So I could say come to this location based on a certain x value and a certain y value, right? And we talked about how you move around the coordinate system in series one when we talked about, you know, just basic Cartesian coordinate system. And if you don't know how to move around the Cartesian coordinate system, it's a non-ASMR math video, but you might want to take a look at that, right? Basically, the way it works is this is our x axis, and x moves along like this. So we're going to go a certain x value here, right? And that's going to take us to this location, that's the x coordinate, and a certain y value here, right? That's our y value there, and that's our x value there. Now the way it works is this just becomes a right angle triangle, right? So let's bring our ruler back or our level back and just draw a right angle triangle, the little thing here, this thing. I got to remember to draw the lines on this side, not go through, and that's what little thing we have right there, right? So what we got right now, so what we got right now, the length, the distance from this point to this point is just our y, right? This is how far we've traveled up to get to this location. So we can just call this the y distance, right? And the distance we've traveled from here to here is just our x coordinate, right? It's just the x distance. And this gives us our right angle triangle, and this is how triangles are related to circles, because what's going to happen right now is if you start moving around the circle, if you start moving around the circle anywhere on the circle, all that's going to happen is we're going to create a right angle triangle, right? That's the way we find ourselves around the circle, and that's the simplest, the quickest way for us to find out where we end up around the circle, right? So if you continue to move around the circle, let's put another guy here. Let's assume there's gravity here, so we're actually upside down here, and as soon as you come around here, you've gone one cycle, one period, right? And that's what we say when it comes to, you know, naming these things. If you go one complete cycle, that's one period. For a clock, you start the cycle here, and 12 gets you to the beginning, right? With the Cartesian coordinate system, with the circle, with the unit circle, it's been standardized with us starting here and on the positive x-axis and going counter-clockwise, okay? Now, so this is our circle and how it relates to the triangle, the right angle triangle, what we'll do, we'll draw another triangle here, and just so you see that nothing changes with this, right? So if we do this, then what we have here is another right angle triangle with the distance of y, and since this direction is negative for the x-value, the distance this way is going to be negative, right? I'm just going to call it x, or you can think about it as negative x if you want, and when we draw this other triangle, there's two different angles we can reference, right? We can reference it from the positive x-axis going this way, and we're going to call this theta 1, I guess, if that's theta, this is theta 1 taking us to the new location, or we can reference it based on this theta, I'm going to call this r, and the reason I'm going to call this r is not arbitrary, it's because the angle which is closest to the x-axis we refer to as the reference angle, right? So if we measure angles from the positive x-axis going counter-clockwise, we call those angles angles in standard position, right? So we're going to write this thing over here, right? So theta s, let's just put a little s down here, right? And we'll call this s1, and subscripts in mathematics are basically like last names, right? They give us more accurate understanding of what it is that we're talking about, right? Whatever your name is, I'm pretty sure there are a lot of other people with your name, your name plus your last name, that narrows it down to fewer people, right? That's what subscripts are in mathematics. When we write a symbol and then put a little thing at the bottom on another little thing at the bottom, all that is, is just us being more accurate with our terminology so we know exactly what it is that we're talking about, okay? So when you're moving around the circle, any angle that you measure from the positive x-axis going counter-clockwise to a length, and these are called the terminal arms, right? So this is called a terminal arm wherever you end up being, right? So if you're standing here, the line from the origin to where you are is called the terminal arm of the circle, right? Wherever you're going. So that's called terminal arm. So you can think of the terminal arm as where you are on the circle, okay? The angle coming off the positive x-axis going counter-clockwise is called the reference angle. It's called the angle in standard position, right? So theta, s, we're going to refer to angle, and now one of the other ways that we can figure out where we are, what the angle is to the terminal arm, is measure the angles from the measure, the angle closest to the x-axis, right? So what we do here is the angle in this quadrant, and this is four different quadrants that we have for the circle, right? This is quadrant one, quadrant two, quadrant three, and quadrant four, right? If we broke it up. So in here, the angle in standard position becomes the reference angle. In here, the angle in standard position is different than the reference angle, right? So the reference angle is just the angle closest to the x-axis, right? So theta is reference angle, and closest, closest to the x-axis, right? Axis. Angle in standard position is from positive x-axis counter-clockwise, right? So that's three words that we have to learn, right? In trigonometry, they're super important. There's one other word that you have to learn related to circles that's super important is coterminal angle, and coterminal angle is basically what other angle gets you to the terminal arm, to the terminal side, right? So when you're moving around the circle, from here, all the way around is 360 degrees, right? So angle in standard position comes off the positive x-axis and goes this way. 360 degrees, you're back to where you were, right? Now, if I say move in the opposite direction, start here and go clockwise, that's considered to be negative, right? So I could tell you to come to this terminal arm, to come to the side, okay, by telling you not to go this way, but to go this way, and that becomes a negative theta, right? It's like the x-axis, right? This way is positive, and this way is negative, right? Angles is the same thing. This way is positive, and this way is negative. So for us, there's multiple ways. There's actually an infinite number of angles which will get us to a certain terminal side, right? Terminal arm, if we had another triangle here. I could tell you to go from here, this way, positive direction with angle, and it gets you to this arm, or I could tell you to go this way, negative direction, it gets you to this arm. I could tell you to come here, and then go another 360, you end up at the same place, and then go another 360, you end up at the same place, right? And this goes an infinite number of times, right? Because it's a cyclic function, it continuously repeats, right? If you think of a wheel going around and around and around, it passes the same point multiple times, right? On your car, you have RPM reps per minute, right? Or whatever the terminology is. That's basically telling you that you're going around a certain number of times per second or per minute or whatever it is. So there's another term that we have to learn when it comes to circles is the angle, but coterminal angle. And coterminal means any other angle that gets you to the same location, okay? And what we're going to do, we're going to call that Theta C. So these are the four words that you really have to learn, because what we're going to do is start talking about the circle, but we're going to talk about the unit circle. And the unit circle basically standardizes things for us. Instead of giving us a radius of nine, what we're going to do, we're going to standardize that and just call the radius of one, okay? Because, well, one is easy to scale, right? So it's really just us making calculations easy for us and simple for us to do. And that's what we're going to talk about. So what we're going to do right now, now that we've talked about the terminology, and this is, again, words that you have to know, right? And if you want an example, let's do an example of this, right? Let's figure out what a certain angle is going to be and find out what the angle and standard position, the reference angle and the coterminal angle are for wherever we're going to end up, right? So let's say we're going to go to this location here where this guy is right now. So we're going to draw our triangle, or we won't draw our triangle. What we're going to do is we're going to draw our terminal, terminal arm, right? Which basically means we ended up there, right? So what we got is we've traveled to this, right? To this location here. We've moved around the circle from here and ended up here. Now, I don't have a protractor. I don't know what that angle is, but I'm going to estimate it. Now, the way you estimate, the way you can do an estimation, and we'll talk a lot more about this. So this is sort of like the introduction of us learning terminology, right? That's what we're mainly concerned about. We'll get into exact calculations in future little segments. So from here, let's assume we traveled to here, okay? Now, if you go halfway around the circle, as we talked about, if you do a 180, you've turned around and gone in the opposite direction, right? So if we're going in this direction, now we're going in this direction, right? We're in opposite ends. That's 180 degrees, a full circle is 360 degrees. And since this is divided into four quadrants, up to here, that's 90 degrees, obviously, right? These two lines, the coordinates are somewhat perpendicular. So if you go from here to here, you've traveled 90 degrees. If you go from here to here, you've traveled 180 degrees. If you go from here all the way to here, that's 390s, right? That's 270 degrees. And if you go 490s, that's 360 degrees. So this angle here, approximately, let's assume it's going to be 225 degrees, okay? So we're going to call this theta in standard position as 225 degrees. Now, the reference angle for this terminal arm is going to be the closest angle to the x-axis, right? And we don't care which axis, which part of the x-axis it is. It sort of acts like a magnet. It gets to the x-axis as quickly as possible. So the reference angle for this terminal arm is this guy right here, right? Because if you're here, you get stuck then towards the x-axis, right? So this is called the reference angle. And the way we figure that out is we go from here to here was 180. From here to here was 225. So all we've got to do is subtract the 180 and that gives us that, right? If we take this part away, 180 minus 225 gives us the reference angle here. And the reference angle here, theta r, is going to be 45 degrees, okay? That's the closest angle to the x-axis. We don't care if it's the positive x-axis or the negative x-axis. As far as coterminal angle goes, we can go this way, right? And this would be negative. So one coterminal angle, theta c, would be 360 minus 225. Let's just do the calculation here, right? 360 degrees minus 225, right? Five you can't take from zero. So that's five and 10. So that becomes five. This becomes three and that becomes one. So a hundred and negative 135 degrees ends at the same terminal arm, right? Ends here as well. So it gets you to the same location. So if you were standing here, if I told you to go negative 135 degrees, you would go this way, right? If I told you to go positive 135 degrees, you would probably end up somewhere here. So that would be the wrong direction. That would be the wrong arm we're trying to get to, right? So that's one coterminal angle. I could give you more. I could say, okay, we've got 225 here. Let's add another 360 to this and go all the way around because if you want to go around the circle to where you began, it's 360 degrees. So we can go 225 plus 360. Let's do another calculation. 260 plus 225. So that's going to be 58 and five. So 585 degrees gets you to this terminal arm, to this location. So we would have gone around here and then gone around again and that becomes 585 degrees and we can continuously do that. We could take negative 135 and subtract 360. And that, again, will end up at the same location, right? So that's sort of an infinite loop that we're talking about there and how that comes into play in different places where we'll talk a lot more about this when we start talking about trick functions when we start graphing trick functions. So that's sort of the way right angle trigonometry is really related to circles which are cyclic functions and that's how we find ourselves around a circle. And these are the terminologies. These are the terms that you really have to learn and you have to become extremely familiar with them because we're going to continuously reference them. So what we're going to do is take these guys down and create another circle but this time we're not going to label the radius as nine units, nine of these squares. I'm going to standardize this and I'm going to call this the radius one because we can easily work with one and we can scale it. And we're going to refer to that thing as the unit circle and that is where we're going to build our understanding of trigonometry. Okay, learn this terminology, think about this for a little bit as moving around the circle and I'll see you guys in the next video. Bye for now.