 So what we're going to do now in this video is take a look at the unit circle and what we're going to do, we're going to take a look at the y value, we're going to take a look at sine theta and see how that varies as theta increases, right? As we move around the circle, what happens to the y value as we move around the circle because the y value is our y part of the of the coordinate, right? Of our Cartesian coordinate system, right? And the cos is just the x value, right? So if we can understand what happens to y as we move around the circle and what happens to x as we move around the circle, the x coordinate, then we've gone one step further in understanding the unit circle, which is basically our ideal cyclic function, right? So let's take this graph and what I'm going to do, I'm going to take this piece of paper off and take one of these pieces of paper from under here and put it on here because we're going to use graph paper, it might as well be as accurate as we can. We're going to use graph paper to graph the sine function, to graph the cos function and to graph the tan function and then hopefully we can, you know, fit it all onto this space right here. And these are going to be curves that, curves that hopefully you've seen before or you're familiar with or, you know, they'll, it's something that you recognize and they're, they're beautiful and they're, they, they come into play in all aspects of society and all aspects of our lives and all aspects of nature, right? Because the light wave, the way you see things, the way light penetrates your eyes, that's just the wave, the way you hear me, the sound that you're hearing right now, those are vibrations and those are basically just waves, right? So if we're able to understand the cyclic function, right? If we're able to understand what happens to us as we move around the circle, what happens to our coordinate system, the X and Y parts of the coordinate system, then we've gone a pretty big step towards understanding how wave function works, right? And the ideal wave functions are basically our sine and cosine functions. So let's take this guy off and keep this in mind, right? You're going to have to know what these are, sine, cosine and tan, so cotoa, right? Sine is opposite, over hypotenuse, cos of an angle is adjacent over hypotenuse, and tan of an angle is opposite over adjacent, and the range, I should put an E on the range, right? And the range of a function is what the possible Y values are, and the domain of a function is what are all the possible X values. So let's just take this off. I don't have the best writing, my very poor writing abilities, and my spelling is not very good. So hopefully when we're talking about it, you know what these are, and maybe you're taking notes yourself, right? So we took this guy down, we're gonna have to access one of these sheets, right? So let me take this down, and maybe I'll be able to pull one off without totally removing this. Okay, now it's not gonna work. So let's take off this guy, and then we'll put it back on again, right? I think that'll hold. So let's do a little adjustment here. So let's put this guy back up again. So we have our unit circle back up here, and I transferred over, switched over the graph paper over here. So what we're about to do is graph starting here at the coordinate one and zero, and us moving around the circle, and we're gonna basically go like this. That's how the angle is going to change, and it goes, you know, you should be familiar with degrees, and just a side note, we will be talking about radians shortly, which is a unit of measure for the angle. It's just the way we measure the angle, which is when you go up in higher level mathematics, it's way more useful than degrees, right? So we will talk about radians later if you're wondering why we haven't talked about it yet. But right now what we want to do, what I want to do is sort of show you how this all works, because we haven't really done any calculations right yet, right? All we're doing right now is getting a full blown picture of what's going on, and learning some of the terminology that we have to learn to be able to understand trigonometry, to be able to understand cyclic functions and triangles, and the trig ratios, right? Which is basically kicking us off into trig functions, which is what we're about to do right now. We're about to graph a trig function, and the first trig function, we're about to graph the sine theta, right? So we're gonna graph ourselves moving around the circle, a unit circle, and we're gonna see what happens to sine theta, what happens to the ratio of the Y versus the hypotenuse. And since the hypotenuse is one, basically what we're doing is we're taking a look at what happens to the Y coordinate of the axes as we move around the circle one cycle, right? So our X axis on this graph is going to be theta, right? And our Y axis is going to be sine theta. So what I'm gonna try to do is I'm gonna try to fit the sine function, the cos function, and the tan function on here. I'm not sure if I'll be able to do all of them. Actually, let's see, go down this far, we can go down that far. I'm pretty sure we'll be able to do all three trig ratios. Well, the first three trig ratios on this graph, and that's what we're gonna do. So let's put our axes here. This here, what have we got? Let's do that. One, two. What we're gonna do, we know as we talked about the range of Y values can only go from negative one to one, right? Because that's where we're functioning right now, right? We can't go beyond lower than negative one and we can't go above one. So if this our x-axis is now going to be theta, okay? And keep in mind, we're doing the graphs, we're doing the mathematics. So it's really up to us what we want to call the x-axis and what we want to call the y-axis, right? That's the power of putting things on a Cartesian cornsus and on a grid or graphing things, right? We can compare anything to anything else and that's what this is. That's what the grid is. We're trying to find out what the relationship between two units are, two variables are, right? And variables just basically something that varies. So what we're gonna do is gonna vary, change theta, and we're gonna find out how the Y part of the coordinate changes, right? So theta is gonna be our independent variable on the x-axis and sine theta is gonna be our dependent variable and it's gonna be dependent on how theta changes, right? So what we're gonna do, we're gonna call the x-axis theta and we're gonna call the y-axis and we're gonna take it 1, 2, 3, 4, 5 and we're gonna call this sine theta. So this guy is gonna be y, which is sine theta, y being our y-axis here, which is sine, right? So let's take a look at this. Let's start off at 0 and 0 for us here means means that we're right here, right? Where theta is 0, this is gonna basically drop down to here, where if we're measuring our angle from the positive x-axis, we're at zero degrees, right? So this is our zero degrees and our y, if we're at zero degrees, just comes down, comes down, comes down, comes down and our y just basically shrinks until it's equal to zero, right? I hope you can see that, right? So our y part of the coordinate is gonna be zero as well. So when we're graphing the sine function, when theta, the angle is zero, sine theta is zero. So what we're gonna do right now is go around and we're not gonna click on every single point here. What we're gonna do is we're gonna go to the four quadrants to the main places where our y value changes because this guy is gonna go from zero, y changes from zero, goes up, goes up, goes up to one and then comes back down again to zero and then goes down to negative one and then comes back up again to zero, right? As you move around the circle. So when we're at 90 degrees, because as we move up here, that's 90 degrees or y becomes one, right? And remember, we're not here, we're not here, we're not here, we have to be on the circle, we can't go off world. So at 90 degrees, so at 90 degrees, I'm just gonna go over four squares here on my grid paper to go to 90 degrees, right? So this is 90 degrees. My y value of the coordinate is one. So at 90 degrees, sine theta is one, right? At 180 degrees, because this is 180 degrees, if we go half, if we go half a circle or a straight line, is 180 degrees. At 180 degrees, my y value comes down to zero again, right? Because this guy just drops, the y part just drops down to zero. So we're going to go one, two, three, four. So at 180 degrees, we're back down to zero. At 270 degrees, we're down here. From the positive x-axis, if we measure it, 270 degrees, we're down here. And my y value is negative one, right? So we're going to go one, two, three, four, and then one, two, three, one, two, three, four. So we're down here. And when we go back, when we go to 360 degrees, we're back where we started, and that was zero, right? So our y value becomes zero. So this was 270 degrees. One, two, three, four. So that's 360 degrees. Oops. 360 degrees. We come back here. Now, the way it works is this isn't straight lines. This doesn't go pch. It doesn't do that, right? We're going to actually create functions that do that. But this doesn't do that right now. And if you ever, you know, if you want to convince yourself, and we will do this later on when we start doing calculations, right? If you want to convince yourself what this picture looks like, you can just grab a calculator and, you know, punch in and take sign of a certain angle and find out what the y coordinate is. But the way it's going to work out is basically this. Because this is curved, right? The y value doesn't go up like a line. It's not going to be a line. It's actually going to be a curve as well. So when you take a sign of, let's say 45 degrees, when you're halfway between, you know, the 90 degrees, if you take sign of 45 degrees, let's do that. Why not? Let's show this to you. So I'm just going to punch in. This is an old calculator. This is from like 1980s. Okay. It's considered to be one of the best solar calculators. One of the first ones, I think, that came out. And they still have it. It went 2015. This was basically my first real calculator that I ever bought. And I guess I was just lucky enough to get my hands on one that was amazing. So what we're going to do, we're going to punch in 45, right? And we're just going to take the sign of it. So sign of 45 degrees, it gives it to you as 0.707. And this is a special triangle. And we'll talk about special triangles and how all that stuff is related to the unit circle and how they come into play in trigonometry. Once we finish covering just an overview of what's going on, because when I teach math, when I teach trigonometry, I really, for the first time that I ever get together with someone, if they're studying trigonometry, I give them a really broad overview. So they have an appreciation for what it is that we're studying and why we're studying this thing. And I hope I'm not going too fast in this. If you're getting lost in this, don't worry too much, because we're going to go through all this again with numbers. We're actually going to do calculations, do problems. Right now, my main just is, my main purpose is to give you a nice broad overview. And then we'll hit up some of the specifics of it, right? With the special triangles and we'll bring radians into play here. So when we take the sign of 45 degrees, we get a value of 0.707. And what that means is when we're 45 degrees here, right? When we're halfway in angle and degrees between here and here, our y value is 0.707. dot dot dot is an irrational number. So we're actually here. We're not halfway here. We're not halfway between 0 and 1. We're further up. We're more than three quarters of the way up here, our y coordinate is more than three quarters of the way up. And if we take cos of 45 degrees, right? So 45, and we're going to take the cos of it, cos of 45 is 0.707 again, right? So when we're halfway between angle wise between this and that, right? Our x coordinate is also three quarters of the way through. We're not at 0.5, right? Which is a little weird. It's not really intuitive. You think if you're halfway between two lines at 45 degrees, you should be halfway on the coordinate system, but you're not. Because this thing arches, right? It's a curve. It's an arc. So it follows an arc. So once we graph us moving around the circle, the y coordinate, this is what it's going to look like. Now that is considered to be one period, one full cycle. And this is a circle. And the first video we took, we did, we talked about coterminal angles and this thing is cyclic and it continues on forever. So we can't, you know, this is, we just graphed one period, one cycle. This thing continues. So this thing goes arrow and arrow, right? So this thing keeps on going and back this way again because, well, the dynamics of it doesn't change. If we move this way, if we move as a negative degrees, negative angle, if we go backwards, well, all that happens is the y is changing in exactly the same way, but going negative first, right? So that's the graph of what the sine function looks like. Let's graph the cos function, okay? And what we're going to see is, it's basically the sine function shifted over 90 degrees because if we take this piece of paper, our unit circle and rotate it, we've got the x up here and the y down here, all it is is going to be basically the same graph. But since our x axis is this way, when we're graphing the cos function, it's just going to be shifted over 90 degrees, right? So what we're going to do is do the same thing. Let's do another graph. One, two, three. So what do we got? One, two, three, four. So one, two, three, four. So I'm just going to draw this thing straight up. So one, two, one, two, three, four. So we started here. So I'm just going to draw this. I'm going to extend this all the way down. That way we don't have to worry about. So that's where my cos is. So I'm going to take it up to here, my y. So that's going to be four. Four, one, two, three, four. So this is going to be my x axis for the cos. So let's graph the cos function. And this is going to be versus theta. Our x axis is again theta, how the angle changes. And this is our independent variable. So what we're going to do is change the angle of the terminal arm, right? Basically move around the circle. And we're going to take a look at what happens with our x value, right? The x part of the coordinate. So that's our theta. And our y axis here now is cos theta. So this is cos theta, right? So we're going to start off here again, right? And we're going to move counterclockwise, right? So we're going to measure angle from standard position, right? So if we're here, our x coordinate is one. And since we're on a unit circle, that means cos theta is one, right? Because cos is adjacent divided by the hypotenuse. And adjacent here is, well, there is no adjacent, right? Is one over one, which is one. So one, two, three, four, one, two, three, four. And this guy's negative one for the sign, right? So again, our angle is going to be theta, three, four, 90 degrees, two, three, four, 180 degrees, three, four, 270 degrees, 360 degrees, right? So when we're at zero degrees, our cos is zero. When we go to 90 degrees, our x value is, sorry, when we're at zero degrees, cos is one, right? So we're up here. When we go to 90 degrees, we're up here. Our cos value is zero, right? Because the x just keeps on getting smaller. This is our triangle. If we're moving along here, x is getting smaller, smaller, smaller, x is now zero, right? So at 90 degrees, x is zero. At 180 degrees, x is negative one. At 270 degrees, x again becomes zero. x increases, increases. As we move around and as we move back, x decreases, right? We go back to zero again and back to one again. So this is what it's going to look like. Curve sort of sucks over there. Should be a better curve, right? This is a curve, not a straight line. So that's what a cos function, cos graph looks like. And if you take this thing and if you shift it over 90 degrees, that's just a sine function, right? And these two functions, basically music that you listen to, is multiple functions, sine functions or cos functions, just layer it on top of each other, right? Which, you know, they give you boxes. They give you different frequencies. You can crunch these things. And we're going to take a look at that stuff, right? We're going to go way more intricate. We're going to vertically expand these things. We're going to change the phase, right? Change the period of these things. We're going to translate them, transpose them and flip them. And we're going to do a whole bunch of things to these functions. But this is our basic trig functions, two of them anyway, sine and cos. What we should do now is just graph the tan function and take a look at what the tan function looks like. And just for those of you who have followed trig and arm tree, there's three other trig identities as well, right? And those are cosecant, secant, cotangent. And those are basically just one over sine, one over cos, one over tan. So we take a look at what happens around a unit circle as we move around, as we move around the unit circle to the inverse of these things, right? To the reciprocal of these things, not the inverse, to the reciprocal of these things. So it gives us different graphs and they are interesting. And they're cool to take a look at. And these things do have physical meanings, right? I know what the sine cos and tan mean, right? The cosecant, secant and cotangent. I never remember what they mean. And I never really teach and explain it. But if you look it up, there's a graph. You can find these beautiful graphs online that they show you the segments where sine, cosine, tangent, and the other tricks, secant, cosecant and cotangent, what their physical meaning is for a unit circle. Okay. So what we're going to do now is take a look at what the tan function looks like. And this is a little bit more difficult to graph because it doesn't, it's not based on just us reading off the coordinate system for a unit circle. It's the ratio of y divided by x, right? y over x. So we start getting asymptotes because one thing that we've talked about in mathematics is we can't divide by zero, right? It becomes undefined. We get infinity. We really, it's a huge obstacle for us in math where we can't divide by zero. We don't know what happens. We can approach an asymptote in a certain direction and see what happens with that asymptote. But we don't know exactly what happens when we divide by zero. And that's one of the pitfalls of one of the limitations, one of the restrictions we have in mathematics. On the flip side, one of the most beautiful properties of zero is it allows us to solve equations. It allows us to factor things and as long as we have multiple things, multiply it together to give us zero, we can split those up and set each one equal to zero, right? Because the only way you can multiply a whole bunch of things to give you a zero is if at least one of them is zero, right? So if we're trying to solve for an equation, if we don't know, you know, which one is zero, we solve for all of them equaling zero because they're all possible solutions, right? And we talked a little bit about this in the language of mathematics in series 3a or 3b, if you're following that stuff. And it's beautiful when we're learning how to factor polynomials specifically, right? So what we're going to do right now is create the grid for the tan function. And the tan function is not like the sine and cosine function. The range, what the tan values can be change, range from negative infinity to infinity for the values. But at certain theta coordinates, we end up getting asymptotes because we end up dividing by zero, right? As we talked about, tan is y divided by x, which is really sine theta divided by cos theta, right? So if we're dividing by cos theta, any place where cos theta is zero, we get an asymptote. We can't divide by zero. So those become our vertical asymptotes. So let's draw this thing and we'll take it from one, two, three, four. So we'll do it here as well. So what we got here is we're again going to look at what happens to us, what the ratio of y to x is, what the ratio of sine versus cosine is as we move around the circle, as we vary theta, right? So our x-axis is theta and our y-axis is now tan theta. We're going to mark these from one to negative one, but that's not the limit of what tan theta can be. We're just doing this to stay on the same scale as the sine and the cos graph, right? And the critical points that we're looking at at 0, 90, 180, 270 and 360 again, right? At these four coordinates, one, two, three, four, right? At these four points. So we're going to throw these on here, zero, one, two, three, four, 270 degrees, one, two, three, four, 360 degrees, okay? So let's take a look at what happens at zero, right? Because that's where we're starting. So at zero degrees, tan theta is sine divided by cos y divided by x. Our y-value is zero, right? Our x-value is one because our radius is one. Zero divided by one is zero. So that's straightforward. Let's go up to 90 degrees at this point right here because that's where things change, right? As we move around the circle, our y-value is increasing and our x-value is decreasing, right? So if our y-value is increasing, our x-value is decreasing, the numerator is increasing and the denominator is decreasing, right? So let's put tan theta here just so you see, oops, tan theta is sine theta divided by cos theta. So as y, the numerator increases, as the top value increases, the bottom number decreases, what happens is we get a line like this and pretty rapidly we see it just going up to infinity because as soon as we hit this point, our x-value is getting smaller and smaller. And if you recall, if you divide by a really small number, then what you end up getting is a huge number, right? So if we're approaching zero, right? Over here, if we're approaching zero, we're dividing by a smaller and smaller number and at the point zero, the number is undefined, but we're infinitely getting close to zero. That means y divided by x, sine divided by cos, is getting infinitely larger. y is positive, x is positive. So we're going to be getting infinitely larger in the positive direction, positive infinity. So what happens is we end up getting an asymptote here because at 90 degrees, cos is zero, right? And sine is approaching one. And you can see this on this graph, right? As we move along, this thing, this is what we're mapping here, right? Our y-value is getting bigger and bigger and bigger, our x-value is getting smaller and smaller and smaller, and it's approaching zero, right? Cos is approaching zero. All of a sudden, we're going to get a division by zero and we'll get an asymptote. So the graphotan function looks like this. Now let's take this to negative 90 as well because what happens with this is, 90, let's extend this at negative 90 degrees. If we go to negative 90, we're going down this way, right? So we're at 270 degrees. Negative 90 degrees is 270 degrees, right? If we're here, our cos is zero again. So again, we have an asymptote. It's negative one divided by zero. We can't do that, right? So we have an asymptote here as well. And what happens is this graph becomes like this. So we have a curve, that's what a tan function looks like. And if we graph this thing for this zone, what we're going to see is when we go to 180 degrees, we have zero divided by negative one. The sine is zero and the cos is negative one. Zero divided by negative one is zero, right? Zero divided by negative one is zero. And 270 degrees is the same as negative 90 degrees. And we know that's an asymptote because we're going to have negative one divided by zero. So this becomes an asymptote as well. So all of a sudden, we have another zone here. And what the graph of this looks like, if you take a look at the function here, in this area, in this zone from 90 to 180 in quadrant two from 90 to 180, the sine is positive, but the cos is negative. So positive divided by negative is negative. So the graph is below the x-axis, so our tan value is negative. Over from 180 to 270, 180 to 270, our x is negative and our y is negative as well, right? Our y is negative and our x is negative. So negative divided by negative, we get positive. And if we were able to, if my paper was long enough, we go another 90 degrees past 360, we would see the same pattern repeating. So at 360 degrees, we're at zero again. It's the same deal, right? This graph looks like this. And that goes up like that until we hit the other asymptote at 360 plus 90, right? 360, 450, I guess, right? So that becomes another asymptote. Now, this is what the trig functions look like, our basic trig identities are basic, not identities, our basic trig functions look like. That's the sine function, the cos function and the tan function. And there's terminology we have to learn on these as well, okay? As far as the terminology goes, it's pretty intuitive. The period of a function is how long it takes for the graph to repeat itself. And a period of a sine graph is 360 degrees, okay? So the period and t represents the period is from zero, right? To 360. So our period for a sine function is 360 degrees. Our period for a cos function is again, 360 degrees, right? But our period for a tan function is 180 degrees because it repeats every had an 180, right? So a period for a tan function is 180 degrees. So that's one of the first things you have to learn as far as terminology goes, the period of a function. The other thing you have to know about this is the amplitude of a function, what the average between the peak and the trough are, right? The amplitude of a function here is equal to one, right? So the amplitude of a sine or a cosine function is just basically the radius of the circle, right? If we increase this radius to two units, our circle becomes bigger. If we decrease the circle, the radius, then what happens is our amplitude gets smaller, right? And the amplitude for a cos function again, it's one. And the tan function goes from negative infinity to positive infinity and there is no amplitude. The amplitude is infinity, right? It's negative infinity. I'm not sure really what they call it. It's just it goes, there is no amplitude. It goes on forever in both directions. And that's some of the basic concepts that you have to understand about trig functions. And we're going to come back to this when we're going to re-graph these with radians. I use degrees usually in general to introduce trigonometry to people because people are familiar with degrees, but degrees becomes useless to us, not useless, but we don't end up using it in higher level mathematics because everything becomes radians and we'll talk about radians most likely in the next video or the next couple of videos because radians is what brings in the magic number pi, the rational number pi. And on a unit circle, just the teaser on a unit circle, the distance that we travel along an arc when we go half a circle is pi. And what a radian is, short form, is if you travel one radian in angle and radian is an angle. So instead of degrees, you go radians, right? If we travel one radian along the circle, that means we've traveled the same distance as the radius along the arc, okay? So if we travel the radius here for unit circle is one, if we travel one unit along the arc of a circle and we stop, that's equivalent to one radian. And the conversion relation I think is approximately 57 degrees, 57.3 degrees is equal to one radians, but we'll confirm that when we start doing conversions and talking about radians. And this is basically our basics of trigonometry, trig identities. And what we're going to do in the next few videos is introduce the concept of radians and take a look at how radians are related to degrees. We're going to look at special triangles. We're going to graph some functions. We're going to look at some problems. We'll look at the ferrous wheel. We'll look at some tied questions. We'll change these guys, right? We'll change their amplitude. We'll do a phase shift. We'll change the frequency. We'll crunch them up and manipulate these things and flip them and see how the graphs of those look like and what happens with the function? What happens to y is equal to sine theta if we put numbers in there, right? If we start manipulating it and multiplying it with things. And this is it. There's a lot of terminology here. Learn the stuff and get a nice feel for what's going on, right? Trigonometry is not just about triangles, but it's about circles in a huge part, right? And it's about cyclic functions and it's about us understanding cyclic functions and our trig ratios are exactly that. They're ratios of the size of a triangle, right angle triangle as we move around the circle.