Understanding Sine and Cosine Concepts

Hello, welcome back. The title of this lesson is called Definition and Meaning of Sine and Cosine. This is part one. This is probably the most important lesson that you could possibly watch for anybody that needs to study essentially any kind of trigonometry, either in the algebra level, at the trig pre-calculus level, certainly on into the calculus level. We're going to peel back the onion and talk about these first trigonometric functions, which are now called sine and cosine. But we're going to go a little deeper. and we're also going to tackle it a little bit different than most books. Usually, most books are going to tell you kind of like what the definition of sine and cosine are, along with the tangent and the other trig functions that we'll talk about later. And then you'll spend a couple of chapters just solving triangles and kind of getting to sort of understand what they are. And then later on, much later on, you typically learn about the concept of a vector, which is something that we're going to be getting into much later. And then right around that time, the light bulb usually goes off and it's like, oh, I understand why we care about sine and cosine. This is what they're for. I do not want you to wait four chapters to understand what sine and cosine are for. So we're going to tackle it a little bit backwards. I'm going to give you the practical meaning behind what sine and cosine really are. Yes, we're going to talk about the definition. Yes, there's an equation involved, but I don't want you to memorize an equation. I want you to, in your bones, understand what sine of an angle is. What cosine of an angle is. Because this is like learning your alphabet. When you're just a little kid, you're learning the alphabet. And then from that, you construct words. And then from that, you build sentences. And from that, you build paragraphs. And then you're writing poetry, okay? You can't write poetry if you don't know the ABCs, or the letters, or the alphabet. So this definition of what the meaning of sine and cosine really is, is foundational. You cannot go farther without really understanding it. So we're gonna spend some time. I'm going to do a lot of talking and a lot of drawing. It's critically important for you to watch all of it, understand it, maybe even watch it two times. Because when I say this is the most important lesson I've taught in probably the last year, it really and truly is. So I want to make sure you understand that. So forget about sine and cosine for a minute. Let's just take a hypothetical scenario. There is a box. There's a box. Let's say it's on the ground right in front of us here. And there's a box. I get down underneath that box, like I'm going to kind of push it up. And I push on that box. Right, I push up on that box with some force. Now this isn't a physics class, but you're smart enough to know what a force is, right? The higher, the more you push, the bigger the number. So we're going to use the concept of force. I'm going to push on this box with a units, five units of force. In physics, you'll learn the unit of force is called a Newton. If you want to think about it, a five pounds of force, fine. But really the unit of force in physics is really called a Newton. It doesn't matter what the unit of a Newton really is, but I'm pushing on this box. with a unit of five units of force, right? But I'm not pushing straight up or straight sideways. I'm pushing up at some angle, right? So if you want to draw a picture, and we always, always, always want to draw pictures, then basically what's going on here is I have some box, okay? I'm going to label it so we all understand, box, okay? Now, I'm terrible at drawing, but I'm basically down underneath this guy, and essentially I'm pushing on this box, right? At some kind of an angle. So what's going to happen here is this box, I'm pushing right in the corner of it like this, and so this box is going to have some force that's going to be pushing in that direction. What's the size of this force? We're going to call it five newtons. Again, I don't care if you understand what the concept of a newton is. It's just a unit of force. Now, you need to start thinking about when you start pushing on things at an angle. Yes, it is, of course, something you're pushing on at an angle. We all know this. But you can think about this force being broken up into two different directions. This force is acting at an angle, but really there's an angle, there's a part of this force, a portion of it is horizontal, right? And a portion of this force can be thought of as going up. So if you can kind of mentally add up the horizontal portion of the force, along with how high or the upward motion of the force, then together they give you the five newtons. In other words, when I push five newtons at some angle, some of my force is going totally parallel to the ground, and some fraction of my force is going straight up and down. I'm just breaking it up into two different directions. So you can think of this two different directions as forming a triangle. So there's the bottom part of the triangle like this, and then there's the other side of the triangle like this. So any triangle I form with a right angle like this is going to be a right angle, so I'm going to put a little right angle symbol. Now, I'm going to write down what the amount of forces in these different directions are, but before I do that, I want to have you recall that there are certain triangles in geometry that are really common for right triangles, right? You gotta remember in the beginning that sine and cosine and tangent, all this stuff right now is only applying to right triangles, triangles with a 90 degree angle. That's a right triangle. There's special triangles, and we learned in the past, we learned some of these special triangles. So I'm gonna put a little divider bar here. There's something called a three. 4, 5, right triangle. And what it means is, if I know the hypotenuse is 5, and I know this is a right triangle, then the other two sides of the triangle, I already know what they are, right? And this triangle isn't quite written or quite drawn exactly correct, but this side would be 3 newtons, and then this side would be 4 newtons, because it's a 3, 4, 5 triangle. How do I know that that's the case? Well, because remember the Pythagorean theorem applies. 5 squared, which is c squared, is equal to a squared plus b squared, so I could say 3 squared plus 4 squared. On the left I get 25, on the right I get 9 plus 16, so I get 25 is equal to 25. What have I proven here? All I'm saying is that a right triangle with the sides of 3 and 4 and the hypotenuse being 5 works out with Pythagorean theorem that that forms a right triangle. So if I know this is a right triangle, and I know the hypotenuse is 5 newtons, that's why I chose 5 newtons, then I automatically know from geometry the other two sides of the triangle. So what does that mean intuitively? What it means is that some of this force is being horizontally directed. How much of this force? Four newtons of it. In fact, if I wanted to, I could kind of draw this triangle having a little arrowhead right here, because I'm pushing to the right with four newtons, and a little arrowhead straight up, pushing up with the amount of three newtons. You see, the five newton force is acting at some angle and so i need to go ahead and write that angle in here so there's some angle i'm not going to tell you what this angle is right now but it's definitely not zero it's some angle and when i angle it exactly at the correct angle i get a three four five triangle the five newton force has been broken up into what we call components the horizontal part of the force is four newtons and the vertical part of the force is three newtons now even though i gave you the size of this triangle i want you to pretend for a minute that you don't really know the sides of the triangle yet. I gave them to you so you can have it in your mind and you know what they are, but just kind of forget for a second that you know them. If you know that the five newton force is acting at some angle up, let's say it wasn't five newtons, let's say it was seven newtons, or ten newtons, or something else, right? Wouldn't it be nice to know how much of that angled force is in the horizontal direction? Wouldn't it be nice to know how much of the angled force is up in the vertical direction? You see, sine and cosine are the tools in trigonometry to take an angled force like this and break it up into the horizontal piece of the force and the vertical piece of the force. I'm going to say that again because it really is the punchline of this entire thing that we're talking about. Yes, we could talk about trigonometry. I could draw triangles. I can teach you what the sine and cosine ratios are and all that stuff, and we're going to get to it. But fundamentally, what do we really use trigonometry for most of the time? It's if I have a force. or any other kind of what we call a vector quantity like velocity or acceleration or force or even other things in physics like magnetic fields and electric fields. Anything with a directed direction like that is called a vector. I always always want to be able to split those arrows up into how much do I have in the horizontal way and how much do I have in the vertical direction because when I break the problem up instead of angling a force like this if I break it up into four newtons horizontally and three newtons vertically, then I can solve the problem in separate directions, which makes it way easier. If I'm throwing a baseball, it's going to be easier to split the motion into a horizontal motion and a vertical motion. All right, so let's back up the truck. Wouldn't it be nice to know if I'm angling a force in general, how much of the force is horizontal and how much of the force is vertical? The answer is yes, it's very useful. And the sine and cosine functions that we're learning about here That is what their job is, is to actually take an angled quantity like this and break it up into horizontal and vertical component pieces. That is what they are for. That's the thrust of what I'm trying to teach you in this lesson. All right. So let's talk about what that is. All right. So what we have then is we have this angle. Obviously, the angle changes. Then how much of this force is in the horizontal and the vertical direction will change as the angle changes. Okay, so what we have do do is we define something called the sine of theta And over on the other board, I'm going to do something similar. What I'm going to do is I'm going to define something called the cosine of theta. But because I don't want to go back and forth, back and forth to this board and this board, I'm going to redraw the triangle over here, just the triangle part, because I want to have a reference on both boards right here. So this is three newtons. This is four newtons. And of course, the actual original force was five newtons, and it was acting at some angle. I've just reduced the picture. I've taken the picture of the box out, and I've just left the picture of the triangle. Notice this triangle doesn't look exactly the same, but the numbers are what's important here, okay? So if I want to split the 5 Newton force into a horizontal piece and a vertical piece I need two different functions One of them is called the sine function and one of them is called the cosine function one of these things tells me how much of this force is in the Horizontal direction and the other of these functions tells us how much of this force exists in the vertical direction That is what sine and cosine are for they are there to break apart things that are at an angle and tell you how much you have in each direction. That's what they're for. So the mathematical definition of what the sine of this angle theta is, is the following. It's the opposite side of this triangle, the opposite, over the hypotenuse. So hyp means hypotenuse. So what it means is it's the opposite side of the triangle. This is the angle, the angle here in question. The opposite side to this angle is the side with three. And the adjacent side to the, I'm sorry, the hypotenuse is the hypotenuse of the triangle. It's the 5. So if I wanted to calculate the sine of this angle here, I'll write it right underneath. I don't know what the angle is, but the sine of this angle is defined to be the different sides of this triangle dividing here. 3 is the opposite side, and the hypotenuse is 5. So it's 3 fifths. This is a number that I can put in my calculator, right? And what this is telling me is that the sine of the angle is really the ratio. of how much of this force exists in the vertical direction divided by how much total force. So you see what the sine function is doing. It's basically saying how much of the force goes up compared to how much of the entire total force do I have. So it's a ratio. That's why these things are called trigonometric ratios. You'll see that all in your books. They're all ratios, sine, cosine, tangent, and there are more trig functions beyond that we'll talk about later. But sine and cosine are the most fundamental important ones because all of the other functions come from sine and cosine. So the sine function literally is the opposite side of the triangle, opposite to this angle divided by the hypotenuse. Great, you can memorize that. But what does it mean? It's just telling me how much of the force goes up compared to how much total force I have, right? Now, if this is what the sine function is, What is the cosine function? The cosine function is the other side of the triangle. It is the ratio of the adjacent side of the hypotenuse divided by... I'm sorry, the adjacent side of the triangle divided by the hypotenuse. So the cosine of this angle, whatever this angle is, is 4 divided by 5. Okay, so again, it's a ratio. It's a ratio of sides of triangles. All of these trigonometric things are ratios of triangle sides. But what it's physically telling me is it's saying how much of this force is in the horizontal direction. How much of this total force is in the horizontal direction divided by how much total force. So it's a ratio. It's like, you know, if I told you in general, if I said, hey, 0.9 of this force is in the vertical direction, you would know that almost all of it's in the vertical direction. If I told you 0.6 of this force is in the horizontal direction, then you would know just a little bit more than half is in the horizontal direction. So when I tell you that the cosine is four fifths, whatever this number comes out to, I could punch it in a calculator and calculate if I want, but it's just a pure fraction. What it's telling me is the the fraction of the total force that exists horizontally, four divided by five, it's very close to one because most of the force actually is in the horizontal direction. Over here, we're saying the sine of this exact same angle in the both cases is three fifths. So it's less than before. And it's telling me how much of the force exists in the vertical direction compared to how much total force I have. It's a ratio, it tells me in general, how much of the force is going up. compared to the total force. This one's telling me how much of the force is going horizontal compared to the total force. Okay, so what this is basically doing is it's another way of looking at it is it's called a projection. So the sine function, the sine of an angle, is basically telling you what the projection is. In a projection, I'm going to go to the y direction. So I'm going to kind of circle this because it's really, really important. And then the cosine is the cosine of theta is basically the projection. Projection, yeah, like this. To the x direction. So I'm going to kind of like box this. There's a lot more I need to say about it, but essentially that's the punchline. The sine function you need to remember in your mind. The sine function is basically telling me how much of the force goes vertical, and the cosine of the function is telling you how much of the force exists horizontal. One of these functions, the sine function, tells you how much is vertical. The other function tells you what ratio of the force is horizontal. So in your mind, you need to start thinking about the sine goes with the y direction. They kind of rhyme. Sine goes with y. Sine goes with y. You've got to remember that. Cosine goes with the x direction. We're going to be doing that over and over. In your mind, you need to remember sine goes with the y direction. Cosine, that goes with the x direction. I'll say it again. Sine, that goes with the y direction. Sine rhymes with y, right? Cosine, that goes with the x direction. You need to remember that. Sine goes with the y direction, the vertical direction. Cosine goes with the x direction, the horizontal direction. All right? So let's go a little bit deeper into this. We've already said the sine of this angle is 3 fifths. So let's go a little bit deeper. Let's go in a calculator and say, well, the sine of this angle, whatever it is, is 3 fifths. We just calculated that. If I grab a calculator and take 3 fifths, I get an answer of 0.6. What this means fundamentally, actually, I think I'm going to do it right to the side so I can save some space here. What this basically means is that 0.6 of the total force is in Y direction. This is what the sign means. Whatever the number you get is, it's telling you how much of the total force in terms of a number from zero to one. Zero means if the sign came out to be zero, there would be none of the force in the Y direction. If the sign came out to be one, all of the force would be in the y direction. But it's really kind of almost in the middle. About 0.6 of it, a little bit more than half of the force is in the y direction, which makes sense. Three compared to five, that's a little bit more than halfway because the half of five is two and a half. Now, what do we get over here, right? For the cosine of this angle, what do we calculate? Four fifths. When we grab four and divide it by five in the calculator, we get 0.8. What does this mean? This means that 0.8 of The total force of the total force is in what direction? Cosine goes with x. It's the x direction. This is the fundamental essence of what sine and cosine is. Sine and cosine, when you put an angle in your calculator and press the sine button or press the cosine button, you're going to get a number that's going to be between negative 1 and positive 1. We'll talk about why you can get negative answers later, OK? basically you're going to get decimals less than one, right? So in this case, the decimal came out to 0.8. Why is it a decimal? It's because you're taking it as a ratio of triangle numbers. The hypotenuse is always the biggest number. So if I take adjacent over hypotenuse, I'm going to get a number less than one. The maximum this could possibly be is equal to one. When I look at this, I take three divided by five. Again, five is the longest side of the triangle, so I'm always going to get a decimal. So whenever I calculate the sine and cosine of an angle, any angle, I'm going to always get a number less than one. You cannot have the sine or the cosine of an angle and press the button on your calculator and get a number bigger than one. You can't because the cosine is something divided by the hypotenuse. The hypotenuse is the longest side of the triangle. So this ratio always is less than one. Same thing for the sine. But the sine ratio tells you what fraction, what percent if you wanna think of it that way, of the force is vertical. Sine goes with Y. Cosine tells you what fraction or what percentage, if you want to think of it as a percent, because this is 80% multiplied by 100, what percent exists in the horizontal direction? So I told you I do a lot of talking. So you can think of it as 0.6, or you can say 60% of the total force exists in the y direction multiplied by 100. 80% of the total force exists in the x direction. Cosine goes with x, sine goes with y. Okay. So here's one of the biggest punchlines I want to pull out of this. I tell students this all the time, and I really want you to remember it because we're going to go through it. Whenever you hear the word sine of theta. I want you to think the triple equal sign means is equal by definition. This is my little terminology. You see it in math books sometimes. I'm being a little loose here, but I really want you to remember that when you see the word sign on an equation, what I want you to think of is I want you to think of a chopping. Whoops, I didn't spell this correctly. Chopping. Chopping function for the y direction. if the length is equal to 1. Now actually, I'm going to hold on for a second. I'm going to write it down. We're going to talk about it. I've got to get both of them down. When you see cosine of some angle, I want you to say that it's in your mind equal by definition to the chopping function, chopping function in quotations, for the x direction. If the length is equal to one, what do I mean by the length is equal to one? What do I mean by chopping function? This is my words. I guarantee you, you pick any algebra book, any calculus book, any trig book, any precalculus book, whatever, you're never gonna see the word chopping function. It's not something that anybody else says, it's what I say. When I tell you sine of an angle, I want you to think that's a chopping function for y. It's a chopping function, you'll see why in a second. When you see cosine of an angle, I want you to think, that's a chopping function for the x direction. Chopping function is really great. What do you think of when you think of a chopping? Anything chopping. I think of a big axe, and I'm chopping a tree down. I'm literally cutting it. I'm shrinking it, because I'm destroying it, cutting it down to a smaller size. You see, what's happening here is this sine came out to be three-fifths, which came out to be a decimal number. The length of the force was really five newtons, but you see the The chopping function in the y direction, which was a sine, came out to be 0.6. That means that if I apply this chopping function to the 5-Newton force, it chops it down by this times 0.6, and so then I get 3 Newtons. So you see the sine function chops the total force, and it makes it so that I get it and chops it down to however much I have in the y direction. I can look at the cosine version, and I can say it's a chopping function for the x direction. The chopping function for this triangle came out to be 0.8. That means if I multiply the total force times 0.8, it chops it down and it gives me the amount of the force in the horizontal direction. Another way of looking at it is I talked about projection to the x direction. I want you to imagine a light, like a flashlight, right above this triangle. Literally, literally, if I built this triangle out of rulers, like a physical thing, and I built the hypotenuse and all that, and I took a light and I shine it down, what's going to happen? it's going to cast a shadow. The shadow directly under this thing is going to be four units long. If I build a ruler five units long and I shine it down and I measure how long the shadow is, I'm going to get a number that is four long, right? Because this thing is at some angle and so the shadow is going to be shorter. If I go over here and build the same ruler at the same angle, but then shine in the this direction, so I'm casting a shadow in this direction, I'm going to, even though I have a five unit long ruler, I'm going to get a length on this side of three newtons. So there's different ways to think about it and I'm talking about it different ways at once so that you can internalize what they mean. The sine of an angle is the projection, as if I shined a flashlight on this hypotenuse and looked at how much of the shadow exists in this direction. That's how much of the y, the y of this force exist in the y direction. And also it's called a chopping function because 0.6 of it exists in the y direction. So when I say it's the chopping function in the y direction, that's what I mean. It's chopping the thing down in the y direction. If the length is equal to 1, what it means is for every 1 newton of force that I have here, 0.6 of it is in the y direction. So if the force were actually not 5 newtons, if it was really 1 newton, then only 0.6 would be. vertical. But if the force over here were 1 newton instead of 5, and I'm looking in the x direction, and the x direction comes out to be a chopping function called a cosine of 0.8, that means that if it were really one unit long, 0.8 of the force would exist in the horizontal direction. And if the force were really one newton long, 0.6 would exist in the vertical direction. So here I'm about to get to the punchline, okay? We said that sine and cosine... are actually trying to break apart these angled forces and tell you how much is horizontal and how much is vertical. The chopping function for the x direction is called a cosine. It came out to be 0.8 in this case. The chopping function for the sine is called the sine function. It goes with the y direction. It came out to be 0.6. So then if I do the following thing, this is the punchline. I'm going to say then, right? If I take the actual hypotenuse of this triangle, 5, And then I multiply by the sine of the angle there, which is, remember, called a chopping function. So I'm chopping, if I can spell chop, okay, I'm chopping the 5 down. I'm chopping it down in the y direction. By how much? Well, by 0.6, because the sine of the angle in this case came out to be 0.6. If you grab a calculator and say 5 times 0.6, what are you going to get? 3 newtons. I have taken the 5. length hypotenuse, the five Newton force, and I have chopped it in the Y direction so that I now know that my force is really three Newtons in the vertical direction, all right? What do I do if I do the same exact thing over here? The chopping function for the X direction came out to be 0.8. So if I take my hypotenuse and then multiply by 0.8, what am I gonna get? Grab a calculator and do this, what you're gonna get is four Newtons. So the three Newtons is in the Y. direction. It's chopped the hypotenuse into the y direction, and this is in the x direction. It's chopped it in the x direction. So here's the big picture. I have a force. It's five newtons angled at some sort of oblique angle. I don't even know what the angle is, but it's some angle up and to the right like this, right? Somehow, even though we knew the sides of this triangle, pretend that you don't. I calculate the sine and the cosine of that angle, and I get what we call a chopping function, a number. It tells you how much of the actual force exists in the horizontal and the vertical directions, but only assuming the length is one is actually one. So that's why I wrote it down here. I said, it's a chopping function for the x direction, assuming the length is one. That means if the force was one newton, then 0.8 of it would exist in the horizontal direction, right? But the length is not one newton. The length is five newtons. So I take this number. And I multiply it by five because this is the chopping function in assuming the length of the triangle, the hypotenuse of it was actually one. But it's actually five times bigger than that. So I take that number and I multiply by the five and I get the four. Now, the sine function is the chopping function for the y direction. It's the projection into the y direction like that. It's a shadow in the y direction like that. OK, it's assuming that the length of the force is actually one. But the length isn't 1, it's 5 times bigger than that. So I take the chopping function and I multiply by 5, and it chops it down and it tells me that 3 newtons of this has to exist in the y direction, and 4 has to exist in the x direction. Now why does this whole thing work? Let me show you why it works. The reason this thing works, why? Because of the following thing. If I take the hypotenuse, which was 5, And I multiply by the sine. What is the sine? If I go back to my definition, the sine was opposite over hypotenuse. Then I multiply by the opposite over the hypotenuse. That is what the sine is. Hypotenuse cancels with hypotenuse, and I'm left with the opposite side. So that is why when I take the hypotenuse and I multiply by the sine, I get the answer in the y direction because the hypotenuse times the sine means the hypotenuse is canceled, and I get back the opposite side, which is in the y direction. If I look at this and I say well the hypotenuse, if I multiply by the cosine, which is what I did right here, what is the definition of cosine? The definition of cosine is the adjacent over the hypotenuse. Adjacent over hypotenuse. The hypotenuse is cancel and I end up with the adjacent side, which is in the x direction. This is all telling you the mathematical reasons why it works. The sine function is defined to be opposite over hypotenuse. It is a ratio of sides. So if I take this ratio and then multiply by the hypotenuse, the hypotenuse is cancelled and it always gives me the opposite side. That's why I say it's like a chopping function. The sine chops the hypotenuse down and gives you the y component of it. The cosine is defined to be the adjacent over the hypotenuse. So if I take this and I multiply by the hypotenuse, the hypotenuse is cancelled and I'm left with the adjacent side. this side on the bottom so that whenever I end up doing that, I get this hypotenuse chopped down into the x direction. I get a shadow. four units long when I look at the projection looking down into the horizontal direction, and I get a shadow three units long when I look at how long the shadow is in the vertical direction. So you can think of it as projections. You can also think of it as chopping. However you want to think about it, that's what it's doing. Sine chops in the y direction, cosine chops in the x direction. All right, I have done a ton of talking, and I have more talking to do, but I think I'm going to be able to cruise along a little bit faster because once I get those ideas out there, you'll be able to follow what I'm going to say next a little bit easier. So this last triangle, I did not know what the angle was, but it was a 3-4-5 triangle. Let's construct a slightly larger triangle now. The angle of that triangle actually works out to be 36.87 degrees. How do I know? I am not going to actually tell you why I know, because it comes later. But let's just say that I actually build this triangle. three, four, five, and I measure the angle with a protractor, I'm actually going to get 36.87 degrees. I know that it's correct because if I actually go into a calculator and I take the sine of 36.87 degrees, like if I put it in degree mode, put this and hit the sine button, I'm going to get something really close to 0.6. If I take the cosine of 36. 87 degrees and hit the cosine button i'm going to get something really close to 0.8. Doesn't that look familiar? The cosine we calculated exactly to be 0.8 and the sine we calculated exactly to be 0.6. So this angle of 36.87 degrees is exactly the angle in this triangle if you measure with the protractor. Now what I'd like to do is let's construct a different triangle. Same angle, exactly the same angle but just physically larger. So it goes way out here I'm pushing, I would be pushing then with more than five newtons of force, and I would have some other y component and some other longer x component, right? So let's construct a larger triangle. Same exact thing, overall shape, but just physically larger triangle. Okay, so let's do that. Now we now know we're making a similar triangle to that, so it's 36.87 degrees, and let's say that this triangle, it was 12 newtons long. in force. So originally I had the box over here and I was pushing at an angle of 36.87 degrees at an amount of force of five newtons. And when I did that, four of the newtons was horizontal and three of the newtons of the force was vertical. Now I'm going to take the same exact box. I'm going to push exactly with the same angle, 36.87 degrees. Everything's the same, but now I'm going to push more than twice as much. Twice as much would be 10 newtons. I'm actually going to push with 12 newtons. Now my question to you is if I push with 12 newtons of force, how much of the force will exist in the horizontal direction? How much of the force will exist in the vertical direction? Remember this is also a right triangle because of the 90 degree angle right here. That's what I want to know. If I push with more force, how much is vertical, how much is horizontal? You know the numbers will be bigger than before, but what are they actually? Well you remember the sine just chops the hypotenuse into the y direction. Sine goes with y. The cosine just chops the hypotenuse into the x direction because cosine goes with x. So then I would say, okay, I'm going to start by saying that the sine of this angle, which I can put it into a calculator, 36.87 degrees, is going to be equal to 0.6. And I know the cosine of this angle, 36.87 degrees, is going to be equal to 0.8. Okay, so how would I calculate what's going on in the y direction? What I would say is that the sine of this angle, the sine is the chopping function in the y direction. So I'll take the actual, I should say, it's the chopping function for the y direction, assuming the force was actually only 1 newton. It's the fraction, it's the amount of the force that would be in the vertical direction if I only push with 1 newton. But I'm not pushing with 1 newton, I'm pushing with 12 times that. So what I'll do is I'll take 12, which is the hypotenuse, and I'll multiply it by the sine of... 36.87 because this is the chopping function in the y direction it chops down this big number into something smaller then what would i get i would have 12 times 0.6 and what would i get there 7.2 newtons so the y direction would be 7.2 newtons that means that this would be 7.2 newtons that would be in the vertical direction if i were to build this thing with a 12 unit long you know piece of wood shine a light at it, I would measure a length over here of 7.2 units tall. Now let's do the exact same calculation here in the x direction. This cosine comes out to 0.8. That means that it's the chopping function if the hypotenuse were only 1. But it's not 1. The hypotenuse is 12 times bigger than that. So because of that, if I want to know how much is in the x direction, I'm just going to take the actual hypotenuse length, and I'm going to multiply it by the cosine of 36.87. And I'm going to get then 12 times 0.8. And that's going to work out to, in the x direction, 9.6. Newtons. That means this direction right here is 9.6 Newtons. So for this triangle, if I push with 12 Newtons at an angle of 36.87, the horizontal amount of force is 9.6 Newtons, and the vertical amount of the force is 7.2 Newtons, right? So I've taken that 12 Newton force and I'm able to figure out using sines and cosines, how much is horizontal, how much is vertical? Because sine chops in the y direction. And cosine chops in the x direction when you then multiply by the hypotenuse. That's what basically is going on here. Now let's verify. Is this correct? Let's verify. Well we know that c squared is a squared plus b squared. So the hypotenuse came out to be 12. So we have 12 squared. a and b are these numbers. So we let's have 7.2 squared, 9.6 squared. Well, 12 squared comes out to 144. 7.2 squared comes out to 51.84. And then this comes out to 96, I'm sorry, 92.16. So when you take 51.84 plus 92.16, you get exactly 144. That means that these sides of these triangles that I calculated really do form a right triangle because Sine and cosine really only applies to, at least now, it only applies to right triangles that have a 90 degree angle in there. That means the Pythagorean theorem must always hold. So when you calculate the sides of these triangles in the end, you must always get sides that always obey Pythagorean theorem. Okay. Last thing I want to say is that the, or for this board anyway, is that the sine and the cosine that you get, of course, we all know it's the ratio of the triangles and all that stuff. but it is really only a function of the angle. The sine that you get when you have a really, really shallow angle, like let's say one degree angle, if you put that in your calculator and get the cosine of it, you're going to get a number that's going to be much higher than 0.8. It's going to be much closer to one because most of the force will be in the x direction. If I have a one degree angle like this and I take the sine of one degree, I'm going to get a number really close to zero because there's very little of that force in the vertical direction if I have an angle really small like this. In fact, let's check these ratios by calculating the sine and the cosine. Remember, the cosine is the adjacent over the hypotenuse, and the sine is the opposite over the hypotenuse. So now that we have figured all this stuff out, let's go over and do that, and I want to do it right below here. The sine of an angle is going to be equal to the opposite over the hypotenuse. That's what the definition, the mathematical definition of the sine is. But in this triangle, the opposite to this angle is 7.2 newtons. The hypotenuse is 12 newtons. So the sine of the angle that we get when we divide 7.2 and divide by 12, we get, what do you think, 0.6. That's what we already know the sine of it is. Okay? And then the cosine of the angle is going to be equal to the adjacent over the hypotenuse. But the adjacent side of this triangle, adjacent to the angle, is 9.6, and then we'll be divided by 12. 9.6 divided by 12. So the cosine of this angle, when you divide those two numbers, you get 0.8. So you see everything is fine. I took the angle, I stuck it in a calculator, I hit the button, I got the sine and the cosine. I use the sine and the cosine to multiply by the hypotenuse and calculate the other sides of the triangle. But once I have the other sides of the triangle, I can run them back through the definition of the sine and the cosine. I'm going to calculate the same numbers because everything is self-consistent like this. All right, so when I take a number and I stick it in the calculator, what the calculator is doing is it's kind of inventing a triangle in memory and it's dividing sides of a triangle to tell you what the sine and cosine are. All right, one more thing I want to talk about. Here we started with a triangle that was this big, five newtons long in the force. Then we stretched the triangle to 12 newtons along in the force at the same exact angle. Now what I want to do is I want to close the circle all the way back and say let's shrink it down. Instead of making the triangle bigger, let's shrink it down and make it where the actual force is not 5 newtons and is not 12 newtons. Let's make the same triangle at the same angle, but with the force only being actually one tiny little newton, one little newton of force. All right, so let's shrink that triangle so that now it's in one newton of force. So what we have there would be a triangle that would be really small. a little baby triangle like this, right? So what we would have is one newton of force acting up at some angle. What angle would it be? Well it would be the same angle, 36.87 degrees. So if I had a little baby triangle like that with a little tiny little force but at the exact same angle, how do I figure out how much of the force is in the horizontal direction and how much is in the vertical direction? I do the same thing. I calculate the sine and the cosine of this angle and then I multiply by the hypotenuse. So let's do it real quick. Now, if you were to stick this in a calculator and hit the sine and cosine button, we're going to get exactly the same numbers. We already know the sine of this angle and the cosine of this angle. Specifically, we know that the sine of 36.87 degrees comes out to 0.6, and we know the cosine of 36.87 degrees comes out to 0.8. Those are the chopping factors. How much does the ratio exist in the vertical direction? what is the ratio, how much exists in the horizontal direction. So for the y direction, what do I do? I'm just going to say that the hypotenuse is 1 times the chopping factor, exactly the same calculation we did before. Remember before it was 12 times the sine, 12 times the cosine. Here I'm going to say 1 times the sine. So what I'm going to get is in the y direction 0.6 newtons. In the x direction, in the x direction or the x component right is going to be equal to the hypotenuse of one times the chopping factor in the x direction which is 0.8. Multiply that out what do you get x direction 0.8 newtons. Okay why do I care about this? Because remember what I told you a long time ago. Let me pull my boards back and try to tie it all together for you. I said I was very careful I said the sine of an angle is the chopping function or the chopping factor that exists for the y direction assuming the length is equal to one. I said that the cosine of an angle is the chopping factor or the chopping function in the x direction that chops the hypotenuse down and tells me how much I have in the x direction assuming the length of the triangle is equal to one. That's why I take the actual hypotenuse of the triangle and I multiply by the chopping factor because if this exists assuming the length is equal to one. but the hypotenuse is five times bigger than that, that chops it down and gives me the actual X value, and it chops it and gives me the value in the Y direction. Now we're going all the way to the end, and let's say let's shrink the triangle all the way down so that it really is only one newton long. If I take the sine and multiply by one, then I have 0.6 newton in the vertical direction. If I get how much of it is in the horizontal direction, I'll take the chopping factor in the X direction, the cosine, multiply by one. and I'll know that 0.8 Newtons is in the horizontal direction. So 0.8 Newtons like this, this is 0.8 Newtons. And over here, this is 0.6 Newtons. So you see what's going on is when I define the sine and the cosine, the sine is gonna be 0.6 divided by one, which means the sine is 0.6. The cosine is gonna be 0.8 divided by one. The cosine is 0.8. So the cosine and the sine. really are the chopping factors assuming the length of the triangle is just equal to one. That's what they're doing. They're saying, hey, your force is really equal to one. This is how much is in the x and in the y directions. Now, if your triangle is anything other than one, which it always is, then you just take those chopping factors and multiply by the hypotenuse, the actual hypotenuse you have. Then you get how much of it is really in the x direction. You can do the same thing in the y direction. Take your chopping factor and multiply by the hypotenuse. to get the actual amount that's for this triangle when it has a length larger than one. Okay? And this is why when I take the cosine and I multiply by the hypotenuse of these other triangles, I get how much is in that direction, which is the projection and so on. Because when I shrink it down like this, that's what the sides actually come out to be. So we have done a ton of So much so that I want to spend here one or two minutes just going through all of it again, because I think it really helps to see it and hear it a few times. Let's say I'm pushing a box at some angle, a length of force of five newtons. I know that a 3-4-5 triangle is special, and it's a right triangle, the sides of a right triangle, so I label it there. The sine is defined to be opposite side from this angle divided by the hypotenuse, whereas the cosine is defined to be the adjacent side divided by the exact same hypotenuse. So in this case I get 3 over 5, the other case I get 4 over 5, and it's literally the ratio of how much is up compared to the total force, and this is the ratio of how much is horizontal compared to the total force. A handy way to think about it is the sine of the angle is the projection to the y direction. The cosine is the projection to the x direction, so sine goes with y. Cosine always goes with x. Always. I want you to remember that. So if we look at the sine in our case, we've got 3 fifths which comes out to a decimal of 0.6. That means that 0.6 of the total force is in the y-direction as a fraction. 0.6 of the total force. Another way of saying that is the sine of 0.6 is called the chopping function or the chopping factor in the y-direction assuming the length is 1. So if your length really isn't 1, just take that ratio and multiply by how long your hypotenuse really is, and you will get how much of it is in the y direction, right? And same thing here. Here's your chopping factor in the x direction. If I take that number, assuming that the length is actually equal to 1, that's how much of it would be in the x direction. But my triangle is 5 times bigger than that, so I multiply by the 5, and then I get the actual amount in the x direction for my triangle. Then we take the exact same triangle, which we now know the angle is 36.87 degrees, And we make it larger so that I'm not pushing with 5 newtons, I'm pushing with 12. And we do the exact same calculation. If I take the chopping factor, which is this, and I multiply by the hypotenuse, I get the amount of force in the y direction, 7.2 newtons. If I take the chopping factor and I multiply by the actual hypotenuse, then I get exactly how much of this force exists in the x direction. Cosine goes with x, sine's the projection in y, cosine's the projection in x. I get these numbers, I run them through that Pythagorean theorem, and I figure out, yes, this does perform a right triangle. And then I actually go and calculate sine and cosine again using the ratios, and I find that the sine and the cosine that I get exactly match what I got from the calculator before. And then we closed out by saying, let's shrink the triangle so that the actual hypotenuse really is only one newton. We do the exact same thing. We take the chopping factor, this, times the hypotenuse. We take the chopping factor in the x direction times the hypotenuse, And we find out that if the hypotenuse is 1, then the y direction has 0.6 newtons, and the x direction has 0.8 newtons. So for every one unit of length of the hypotenuse, this is the amount, the cosine is the amount of that force in the x direction, and this 0.6 is the amount of the force in the y direction for every one unit of force that you have. That's why when you multiply those chopping factors times the hypotenuse, that gives you the whole projection in those directions. So, I really encourage you to watch this two times. It's a lot, and it's easy to look at it and say, oh yeah, yeah, I get it. But what's gonna happen is we're going to introduce so many new concepts, and calculating different sides of triangles, and then you're gonna get into more advanced classes, and do things with vectors, and all this stuff, and then maybe three months from now, you might say, oh, I get it. I know why sine is like that. I know why sine goes with the y direction. I know why cosine goes with the x direction. I'm trying to bring this up to the beginning. so you know the point of it. Because when you're solving a problem and you're trying to throw a baseball or send a probe to Jupiter or whatever, you want to take the curved trajectory, you want to split it into different directions so that you can analyze the different directions separately. So when we know that we have an angle here with some kind of number like a force, we want to split the directions to see what happens in those different directions. The rest of trigonometry and precalculus and a lot of calculus really is built on this fundamental idea. So watch this several times, go through these calculations yourself, and then follow me on to the next lesson. We're going to get a lot more practice with the fundamental meaning of sine and cosine.

But we're going to go a little deeper. and we're also going to tackle it a little bit different than most books. Usually, most books are going to tell you kind of like what the definition of sine and cosine are, along with the tangent and the other trig functions that we'll talk about later.

And then you'll spend a couple of chapters just solving triangles and kind of getting to sort of understand what they are. And then later on, much later on, you typically learn about the concept of a vector, which is something that we're going to be getting into much later. And then right around that time, the light bulb usually goes off and it's like, oh, I understand why we care about sine and cosine. This is what they're for. I do not want you to wait four chapters to understand what sine and cosine are for.

So we're going to tackle it a little bit backwards. I'm going to give you the practical meaning behind what sine and cosine really are. Yes, we're going to talk about the definition.

Yes, there's an equation involved, but I don't want you to memorize an equation. I want you to, in your bones, understand what sine of an angle is. What cosine of an angle is. Because this is like learning your alphabet.

When you're just a little kid, you're learning the alphabet. And then from that, you construct words. And then from that, you build sentences. And from that, you build paragraphs. And then you're writing poetry, okay?

You can't write poetry if you don't know the ABCs, or the letters, or the alphabet. So this definition of what the meaning of sine and cosine really is, is foundational. You cannot go farther without really understanding it. So we're gonna spend some time.

I'm going to do a lot of talking and a lot of drawing. It's critically important for you to watch all of it, understand it, maybe even watch it two times. Because when I say this is the most important lesson I've taught in probably the last year, it really and truly is.

So I want to make sure you understand that. So forget about sine and cosine for a minute. Let's just take a hypothetical scenario.

There is a box. There's a box. Let's say it's on the ground right in front of us here. And there's a box. I get down underneath that box, like I'm going to kind of push it up.

And I push on that box. Right, I push up on that box with some force. Now this isn't a physics class, but you're smart enough to know what a force is, right?

The higher, the more you push, the bigger the number. So we're going to use the concept of force. I'm going to push on this box with a units, five units of force. In physics, you'll learn the unit of force is called a Newton. If you want to think about it, a five pounds of force, fine.

But really the unit of force in physics is really called a Newton. It doesn't matter what the unit of a Newton really is, but I'm pushing on this box. with a unit of five units of force, right? But I'm not pushing straight up or straight sideways.

I'm pushing up at some angle, right? So if you want to draw a picture, and we always, always, always want to draw pictures, then basically what's going on here is I have some box, okay? I'm going to label it so we all understand, box, okay?

Now, I'm terrible at drawing, but I'm basically down underneath this guy, and essentially I'm pushing on this box, right? At some kind of an angle. So what's going to happen here is this box, I'm pushing right in the corner of it like this, and so this box is going to have some force that's going to be pushing in that direction.

What's the size of this force? We're going to call it five newtons. Again, I don't care if you understand what the concept of a newton is.

It's just a unit of force. Now, you need to start thinking about when you start pushing on things at an angle. Yes, it is, of course, something you're pushing on at an angle.

We all know this. But you can think about this force being broken up into two different directions. This force is acting at an angle, but really there's an angle, there's a part of this force, a portion of it is horizontal, right?

And a portion of this force can be thought of as going up. So if you can kind of mentally add up the horizontal portion of the force, along with how high or the upward motion of the force, then together they give you the five newtons. In other words, when I push five newtons at some angle, some of my force is going totally parallel to the ground, and some fraction of my force is going straight up and down.

I'm just breaking it up into two different directions. So you can think of this two different directions as forming a triangle. So there's the bottom part of the triangle like this, and then there's the other side of the triangle like this. So any triangle I form with a right angle like this is going to be a right angle, so I'm going to put a little right angle symbol.

Now, I'm going to write down what the amount of forces in these different directions are, but before I do that, I want to have you recall that there are certain triangles in geometry that are really common for right triangles, right? You gotta remember in the beginning that sine and cosine and tangent, all this stuff right now is only applying to right triangles, triangles with a 90 degree angle. That's a right triangle. There's special triangles, and we learned in the past, we learned some of these special triangles.

So I'm gonna put a little divider bar here. There's something called a three. 4, 5, right triangle.

And what it means is, if I know the hypotenuse is 5, and I know this is a right triangle, then the other two sides of the triangle, I already know what they are, right? And this triangle isn't quite written or quite drawn exactly correct, but this side would be 3 newtons, and then this side would be 4 newtons, because it's a 3, 4, 5 triangle. How do I know that that's the case?

Well, because remember the Pythagorean theorem applies. 5 squared, which is c squared, is equal to a squared plus b squared, so I could say 3 squared plus 4 squared. On the left I get 25, on the right I get 9 plus 16, so I get 25 is equal to 25. What have I proven here? All I'm saying is that a right triangle with the sides of 3 and 4 and the hypotenuse being 5 works out with Pythagorean theorem that that forms a right triangle.

So if I know this is a right triangle, and I know the hypotenuse is 5 newtons, that's why I chose 5 newtons, then I automatically know from geometry the other two sides of the triangle. So what does that mean intuitively? What it means is that some of this force is being horizontally directed. How much of this force?

Four newtons of it. In fact, if I wanted to, I could kind of draw this triangle having a little arrowhead right here, because I'm pushing to the right with four newtons, and a little arrowhead straight up, pushing up with the amount of three newtons. You see, the five newton force is acting at some angle and so i need to go ahead and write that angle in here so there's some angle i'm not going to tell you what this angle is right now but it's definitely not zero it's some angle and when i angle it exactly at the correct angle i get a three four five triangle the five newton force has been broken up into what we call components the horizontal part of the force is four newtons and the vertical part of the force is three newtons now even though i gave you the size of this triangle i want you to pretend for a minute that you don't really know the sides of the triangle yet. I gave them to you so you can have it in your mind and you know what they are, but just kind of forget for a second that you know them.

If you know that the five newton force is acting at some angle up, let's say it wasn't five newtons, let's say it was seven newtons, or ten newtons, or something else, right? Wouldn't it be nice to know how much of that angled force is in the horizontal direction? Wouldn't it be nice to know how much of the angled force is up in the vertical direction?

You see, sine and cosine are the tools in trigonometry to take an angled force like this and break it up into the horizontal piece of the force and the vertical piece of the force. I'm going to say that again because it really is the punchline of this entire thing that we're talking about. Yes, we could talk about trigonometry.

I could draw triangles. I can teach you what the sine and cosine ratios are and all that stuff, and we're going to get to it. But fundamentally, what do we really use trigonometry for most of the time?

It's if I have a force. or any other kind of what we call a vector quantity like velocity or acceleration or force or even other things in physics like magnetic fields and electric fields. Anything with a directed direction like that is called a vector.

I always always want to be able to split those arrows up into how much do I have in the horizontal way and how much do I have in the vertical direction because when I break the problem up instead of angling a force like this if I break it up into four newtons horizontally and three newtons vertically, then I can solve the problem in separate directions, which makes it way easier. If I'm throwing a baseball, it's going to be easier to split the motion into a horizontal motion and a vertical motion. All right, so let's back up the truck.

Wouldn't it be nice to know if I'm angling a force in general, how much of the force is horizontal and how much of the force is vertical? The answer is yes, it's very useful. And the sine and cosine functions that we're learning about here That is what their job is, is to actually take an angled quantity like this and break it up into horizontal and vertical component pieces.

That is what they are for. That's the thrust of what I'm trying to teach you in this lesson. All right. So let's talk about what that is. All right.

So what we have then is we have this angle. Obviously, the angle changes. Then how much of this force is in the horizontal and the vertical direction will change as the angle changes. Okay, so what we have do do is we define something called the sine of theta And over on the other board, I'm going to do something similar. What I'm going to do is I'm going to define something called the cosine of theta.

But because I don't want to go back and forth, back and forth to this board and this board, I'm going to redraw the triangle over here, just the triangle part, because I want to have a reference on both boards right here. So this is three newtons. This is four newtons.

And of course, the actual original force was five newtons, and it was acting at some angle. I've just reduced the picture. I've taken the picture of the box out, and I've just left the picture of the triangle.

Notice this triangle doesn't look exactly the same, but the numbers are what's important here, okay? So if I want to split the 5 Newton force into a horizontal piece and a vertical piece I need two different functions One of them is called the sine function and one of them is called the cosine function one of these things tells me how much of this force is in the Horizontal direction and the other of these functions tells us how much of this force exists in the vertical direction That is what sine and cosine are for they are there to break apart things that are at an angle and tell you how much you have in each direction. That's what they're for.

So the mathematical definition of what the sine of this angle theta is, is the following. It's the opposite side of this triangle, the opposite, over the hypotenuse. So hyp means hypotenuse.

So what it means is it's the opposite side of the triangle. This is the angle, the angle here in question. The opposite side to this angle is the side with three.

And the adjacent side to the, I'm sorry, the hypotenuse is the hypotenuse of the triangle. It's the 5. So if I wanted to calculate the sine of this angle here, I'll write it right underneath. I don't know what the angle is, but the sine of this angle is defined to be the different sides of this triangle dividing here. 3 is the opposite side, and the hypotenuse is 5. So it's 3 fifths.

This is a number that I can put in my calculator, right? And what this is telling me is that the sine of the angle is really the ratio. of how much of this force exists in the vertical direction divided by how much total force.

So you see what the sine function is doing. It's basically saying how much of the force goes up compared to how much of the entire total force do I have. So it's a ratio. That's why these things are called trigonometric ratios. You'll see that all in your books.

They're all ratios, sine, cosine, tangent, and there are more trig functions beyond that we'll talk about later. But sine and cosine are the most fundamental important ones because all of the other functions come from sine and cosine. So the sine function literally is the opposite side of the triangle, opposite to this angle divided by the hypotenuse. Great, you can memorize that. But what does it mean?

It's just telling me how much of the force goes up compared to how much total force I have, right? Now, if this is what the sine function is, What is the cosine function? The cosine function is the other side of the triangle.

It is the ratio of the adjacent side of the hypotenuse divided by... I'm sorry, the adjacent side of the triangle divided by the hypotenuse. So the cosine of this angle, whatever this angle is, is 4 divided by 5. Okay, so again, it's a ratio. It's a ratio of sides of triangles.

All of these trigonometric things are ratios of triangle sides. But what it's physically telling me is it's saying how much of this force is in the horizontal direction. How much of this total force is in the horizontal direction divided by how much total force.

So it's a ratio. It's like, you know, if I told you in general, if I said, hey, 0.9 of this force is in the vertical direction, you would know that almost all of it's in the vertical direction. If I told you 0.6 of this force is in the horizontal direction, then you would know just a little bit more than half is in the horizontal direction.

So when I tell you that the cosine is four fifths, whatever this number comes out to, I could punch it in a calculator and calculate if I want, but it's just a pure fraction. What it's telling me is the the fraction of the total force that exists horizontally, four divided by five, it's very close to one because most of the force actually is in the horizontal direction. Over here, we're saying the sine of this exact same angle in the both cases is three fifths.

So it's less than before. And it's telling me how much of the force exists in the vertical direction compared to how much total force I have. It's a ratio, it tells me in general, how much of the force is going up.

compared to the total force. This one's telling me how much of the force is going horizontal compared to the total force. Okay, so what this is basically doing is it's another way of looking at it is it's called a projection. So the sine function, the sine of an angle, is basically telling you what the projection is.

In a projection, I'm going to go to the y direction. So I'm going to kind of circle this because it's really, really important. And then the cosine is the cosine of theta is basically the projection. Projection, yeah, like this.

To the x direction. So I'm going to kind of like box this. There's a lot more I need to say about it, but essentially that's the punchline.

The sine function you need to remember in your mind. The sine function is basically telling me how much of the force goes vertical, and the cosine of the function is telling you how much of the force exists horizontal. One of these functions, the sine function, tells you how much is vertical.

The other function tells you what ratio of the force is horizontal. So in your mind, you need to start thinking about the sine goes with the y direction. They kind of rhyme. Sine goes with y.

Sine goes with y. You've got to remember that. Cosine goes with the x direction.

We're going to be doing that over and over. In your mind, you need to remember sine goes with the y direction. Cosine, that goes with the x direction. I'll say it again.

Sine, that goes with the y direction. Sine rhymes with y, right? Cosine, that goes with the x direction. You need to remember that. Sine goes with the y direction, the vertical direction.

Cosine goes with the x direction, the horizontal direction. All right? So let's go a little bit deeper into this.

We've already said the sine of this angle is 3 fifths. So let's go a little bit deeper. Let's go in a calculator and say, well, the sine of this angle, whatever it is, is 3 fifths.

We just calculated that. If I grab a calculator and take 3 fifths, I get an answer of 0.6. What this means fundamentally, actually, I think I'm going to do it right to the side so I can save some space here. What this basically means is that 0.6 of the total force is in Y direction.

This is what the sign means. Whatever the number you get is, it's telling you how much of the total force in terms of a number from zero to one. Zero means if the sign came out to be zero, there would be none of the force in the Y direction.

If the sign came out to be one, all of the force would be in the y direction. But it's really kind of almost in the middle. About 0.6 of it, a little bit more than half of the force is in the y direction, which makes sense. Three compared to five, that's a little bit more than halfway because the half of five is two and a half. Now, what do we get over here, right?

For the cosine of this angle, what do we calculate? Four fifths. When we grab four and divide it by five in the calculator, we get 0.8.

What does this mean? This means that 0.8 of The total force of the total force is in what direction? Cosine goes with x.

It's the x direction. This is the fundamental essence of what sine and cosine is. Sine and cosine, when you put an angle in your calculator and press the sine button or press the cosine button, you're going to get a number that's going to be between negative 1 and positive 1. We'll talk about why you can get negative answers later, OK? basically you're going to get decimals less than one, right?

So in this case, the decimal came out to 0.8. Why is it a decimal? It's because you're taking it as a ratio of triangle numbers.

The hypotenuse is always the biggest number. So if I take adjacent over hypotenuse, I'm going to get a number less than one. The maximum this could possibly be is equal to one.

When I look at this, I take three divided by five. Again, five is the longest side of the triangle, so I'm always going to get a decimal. So whenever I calculate the sine and cosine of an angle, any angle, I'm going to always get a number less than one.

You cannot have the sine or the cosine of an angle and press the button on your calculator and get a number bigger than one. You can't because the cosine is something divided by the hypotenuse. The hypotenuse is the longest side of the triangle. So this ratio always is less than one.

Same thing for the sine. But the sine ratio tells you what fraction, what percent if you wanna think of it that way, of the force is vertical. Sine goes with Y. Cosine tells you what fraction or what percentage, if you want to think of it as a percent, because this is 80% multiplied by 100, what percent exists in the horizontal direction?

So I told you I do a lot of talking. So you can think of it as 0.6, or you can say 60% of the total force exists in the y direction multiplied by 100. 80% of the total force exists in the x direction. Cosine goes with x, sine goes with y. Okay.

So here's one of the biggest punchlines I want to pull out of this. I tell students this all the time, and I really want you to remember it because we're going to go through it. Whenever you hear the word sine of theta. I want you to think the triple equal sign means is equal by definition.

This is my little terminology. You see it in math books sometimes. I'm being a little loose here, but I really want you to remember that when you see the word sign on an equation, what I want you to think of is I want you to think of a chopping.

Whoops, I didn't spell this correctly. Chopping. Chopping function for the y direction. if the length is equal to 1. Now actually, I'm going to hold on for a second.

I'm going to write it down. We're going to talk about it. I've got to get both of them down.

When you see cosine of some angle, I want you to say that it's in your mind equal by definition to the chopping function, chopping function in quotations, for the x direction. If the length is equal to one, what do I mean by the length is equal to one? What do I mean by chopping function? This is my words. I guarantee you, you pick any algebra book, any calculus book, any trig book, any precalculus book, whatever, you're never gonna see the word chopping function.

It's not something that anybody else says, it's what I say. When I tell you sine of an angle, I want you to think that's a chopping function for y. It's a chopping function, you'll see why in a second. When you see cosine of an angle, I want you to think, that's a chopping function for the x direction.

Chopping function is really great. What do you think of when you think of a chopping? Anything chopping.

I think of a big axe, and I'm chopping a tree down. I'm literally cutting it. I'm shrinking it, because I'm destroying it, cutting it down to a smaller size. You see, what's happening here is this sine came out to be three-fifths, which came out to be a decimal number.

The length of the force was really five newtons, but you see the The chopping function in the y direction, which was a sine, came out to be 0.6. That means that if I apply this chopping function to the 5-Newton force, it chops it down by this times 0.6, and so then I get 3 Newtons. So you see the sine function chops the total force, and it makes it so that I get it and chops it down to however much I have in the y direction. I can look at the cosine version, and I can say it's a chopping function for the x direction.

The chopping function for this triangle came out to be 0.8. That means if I multiply the total force times 0.8, it chops it down and it gives me the amount of the force in the horizontal direction. Another way of looking at it is I talked about projection to the x direction.

I want you to imagine a light, like a flashlight, right above this triangle. Literally, literally, if I built this triangle out of rulers, like a physical thing, and I built the hypotenuse and all that, and I took a light and I shine it down, what's going to happen? it's going to cast a shadow. The shadow directly under this thing is going to be four units long. If I build a ruler five units long and I shine it down and I measure how long the shadow is, I'm going to get a number that is four long, right?

Because this thing is at some angle and so the shadow is going to be shorter. If I go over here and build the same ruler at the same angle, but then shine in the this direction, so I'm casting a shadow in this direction, I'm going to, even though I have a five unit long ruler, I'm going to get a length on this side of three newtons. So there's different ways to think about it and I'm talking about it different ways at once so that you can internalize what they mean. The sine of an angle is the projection, as if I shined a flashlight on this hypotenuse and looked at how much of the shadow exists in this direction. That's how much of the y, the y of this force exist in the y direction.

And also it's called a chopping function because 0.6 of it exists in the y direction. So when I say it's the chopping function in the y direction, that's what I mean. It's chopping the thing down in the y direction. If the length is equal to 1, what it means is for every 1 newton of force that I have here, 0.6 of it is in the y direction. So if the force were actually not 5 newtons, if it was really 1 newton, then only 0.6 would be.

vertical. But if the force over here were 1 newton instead of 5, and I'm looking in the x direction, and the x direction comes out to be a chopping function called a cosine of 0.8, that means that if it were really one unit long, 0.8 of the force would exist in the horizontal direction. And if the force were really one newton long, 0.6 would exist in the vertical direction.

So here I'm about to get to the punchline, okay? We said that sine and cosine... are actually trying to break apart these angled forces and tell you how much is horizontal and how much is vertical.

The chopping function for the x direction is called a cosine. It came out to be 0.8 in this case. The chopping function for the sine is called the sine function. It goes with the y direction.

It came out to be 0.6. So then if I do the following thing, this is the punchline. I'm going to say then, right?

If I take the actual hypotenuse of this triangle, 5, And then I multiply by the sine of the angle there, which is, remember, called a chopping function. So I'm chopping, if I can spell chop, okay, I'm chopping the 5 down. I'm chopping it down in the y direction.

By how much? Well, by 0.6, because the sine of the angle in this case came out to be 0.6. If you grab a calculator and say 5 times 0.6, what are you going to get?

3 newtons. I have taken the 5. length hypotenuse, the five Newton force, and I have chopped it in the Y direction so that I now know that my force is really three Newtons in the vertical direction, all right? What do I do if I do the same exact thing over here?

The chopping function for the X direction came out to be 0.8. So if I take my hypotenuse and then multiply by 0.8, what am I gonna get? Grab a calculator and do this, what you're gonna get is four Newtons.

So the three Newtons is in the Y. direction. It's chopped the hypotenuse into the y direction, and this is in the x direction.

It's chopped it in the x direction. So here's the big picture. I have a force.

It's five newtons angled at some sort of oblique angle. I don't even know what the angle is, but it's some angle up and to the right like this, right? Somehow, even though we knew the sides of this triangle, pretend that you don't. I calculate the sine and the cosine of that angle, and I get what we call a chopping function, a number. It tells you how much of the actual force exists in the horizontal and the vertical directions, but only assuming the length is one is actually one.

So that's why I wrote it down here. I said, it's a chopping function for the x direction, assuming the length is one. That means if the force was one newton, then 0.8 of it would exist in the horizontal direction, right?

But the length is not one newton. The length is five newtons. So I take this number. And I multiply it by five because this is the chopping function in assuming the length of the triangle, the hypotenuse of it was actually one.

But it's actually five times bigger than that. So I take that number and I multiply by the five and I get the four. Now, the sine function is the chopping function for the y direction.

It's the projection into the y direction like that. It's a shadow in the y direction like that. OK, it's assuming that the length of the force is actually one. But the length isn't 1, it's 5 times bigger than that. So I take the chopping function and I multiply by 5, and it chops it down and it tells me that 3 newtons of this has to exist in the y direction, and 4 has to exist in the x direction.

Now why does this whole thing work? Let me show you why it works. The reason this thing works, why? Because of the following thing.

If I take the hypotenuse, which was 5, And I multiply by the sine. What is the sine? If I go back to my definition, the sine was opposite over hypotenuse.

Then I multiply by the opposite over the hypotenuse. That is what the sine is. Hypotenuse cancels with hypotenuse, and I'm left with the opposite side.

So that is why when I take the hypotenuse and I multiply by the sine, I get the answer in the y direction because the hypotenuse times the sine means the hypotenuse is canceled, and I get back the opposite side, which is in the y direction. If I look at this and I say well the hypotenuse, if I multiply by the cosine, which is what I did right here, what is the definition of cosine? The definition of cosine is the adjacent over the hypotenuse.

Adjacent over hypotenuse. The hypotenuse is cancel and I end up with the adjacent side, which is in the x direction. This is all telling you the mathematical reasons why it works. The sine function is defined to be opposite over hypotenuse.

It is a ratio of sides. So if I take this ratio and then multiply by the hypotenuse, the hypotenuse is cancelled and it always gives me the opposite side. That's why I say it's like a chopping function. The sine chops the hypotenuse down and gives you the y component of it. The cosine is defined to be the adjacent over the hypotenuse.

So if I take this and I multiply by the hypotenuse, the hypotenuse is cancelled and I'm left with the adjacent side. this side on the bottom so that whenever I end up doing that, I get this hypotenuse chopped down into the x direction. I get a shadow. four units long when I look at the projection looking down into the horizontal direction, and I get a shadow three units long when I look at how long the shadow is in the vertical direction.

So you can think of it as projections. You can also think of it as chopping. However you want to think about it, that's what it's doing. Sine chops in the y direction, cosine chops in the x direction.

All right, I have done a ton of talking, and I have more talking to do, but I think I'm going to be able to cruise along a little bit faster because once I get those ideas out there, you'll be able to follow what I'm going to say next a little bit easier. So this last triangle, I did not know what the angle was, but it was a 3-4-5 triangle. Let's construct a slightly larger triangle now. The angle of that triangle actually works out to be 36.87 degrees.

How do I know? I am not going to actually tell you why I know, because it comes later. But let's just say that I actually build this triangle. three, four, five, and I measure the angle with a protractor, I'm actually going to get 36.87 degrees. I know that it's correct because if I actually go into a calculator and I take the sine of 36.87 degrees, like if I put it in degree mode, put this and hit the sine button, I'm going to get something really close to 0.6.

If I take the cosine of 36. 87 degrees and hit the cosine button i'm going to get something really close to 0.8. Doesn't that look familiar? The cosine we calculated exactly to be 0.8 and the sine we calculated exactly to be 0.6.

So this angle of 36.87 degrees is exactly the angle in this triangle if you measure with the protractor. Now what I'd like to do is let's construct a different triangle. Same angle, exactly the same angle but just physically larger.

So it goes way out here I'm pushing, I would be pushing then with more than five newtons of force, and I would have some other y component and some other longer x component, right? So let's construct a larger triangle. Same exact thing, overall shape, but just physically larger triangle.

Okay, so let's do that. Now we now know we're making a similar triangle to that, so it's 36.87 degrees, and let's say that this triangle, it was 12 newtons long. in force.

So originally I had the box over here and I was pushing at an angle of 36.87 degrees at an amount of force of five newtons. And when I did that, four of the newtons was horizontal and three of the newtons of the force was vertical. Now I'm going to take the same exact box. I'm going to push exactly with the same angle, 36.87 degrees.

Everything's the same, but now I'm going to push more than twice as much. Twice as much would be 10 newtons. I'm actually going to push with 12 newtons. Now my question to you is if I push with 12 newtons of force, how much of the force will exist in the horizontal direction?

How much of the force will exist in the vertical direction? Remember this is also a right triangle because of the 90 degree angle right here. That's what I want to know. If I push with more force, how much is vertical, how much is horizontal?

You know the numbers will be bigger than before, but what are they actually? Well you remember the sine just chops the hypotenuse into the y direction. Sine goes with y. The cosine just chops the hypotenuse into the x direction because cosine goes with x.

So then I would say, okay, I'm going to start by saying that the sine of this angle, which I can put it into a calculator, 36.87 degrees, is going to be equal to 0.6. And I know the cosine of this angle, 36.87 degrees, is going to be equal to 0.8. Okay, so how would I calculate what's going on in the y direction?

What I would say is that the sine of this angle, the sine is the chopping function in the y direction. So I'll take the actual, I should say, it's the chopping function for the y direction, assuming the force was actually only 1 newton. It's the fraction, it's the amount of the force that would be in the vertical direction if I only push with 1 newton. But I'm not pushing with 1 newton, I'm pushing with 12 times that. So what I'll do is I'll take 12, which is the hypotenuse, and I'll multiply it by the sine of...

36.87 because this is the chopping function in the y direction it chops down this big number into something smaller then what would i get i would have 12 times 0.6 and what would i get there 7.2 newtons so the y direction would be 7.2 newtons that means that this would be 7.2 newtons that would be in the vertical direction if i were to build this thing with a 12 unit long you know piece of wood shine a light at it, I would measure a length over here of 7.2 units tall. Now let's do the exact same calculation here in the x direction. This cosine comes out to 0.8. That means that it's the chopping function if the hypotenuse were only 1. But it's not 1. The hypotenuse is 12 times bigger than that.

So because of that, if I want to know how much is in the x direction, I'm just going to take the actual hypotenuse length, and I'm going to multiply it by the cosine of 36.87. And I'm going to get then 12 times 0.8. And that's going to work out to, in the x direction, 9.6.

Newtons. That means this direction right here is 9.6 Newtons. So for this triangle, if I push with 12 Newtons at an angle of 36.87, the horizontal amount of force is 9.6 Newtons, and the vertical amount of the force is 7.2 Newtons, right? So I've taken that 12 Newton force and I'm able to figure out using sines and cosines, how much is horizontal, how much is vertical?

Because sine chops in the y direction. And cosine chops in the x direction when you then multiply by the hypotenuse. That's what basically is going on here.

Now let's verify. Is this correct? Let's verify.

Well we know that c squared is a squared plus b squared. So the hypotenuse came out to be 12. So we have 12 squared. a and b are these numbers.

So we let's have 7.2 squared, 9.6 squared. Well, 12 squared comes out to 144. 7.2 squared comes out to 51.84. And then this comes out to 96, I'm sorry, 92.16. So when you take 51.84 plus 92.16, you get exactly 144. That means that these sides of these triangles that I calculated really do form a right triangle because Sine and cosine really only applies to, at least now, it only applies to right triangles that have a 90 degree angle in there. That means the Pythagorean theorem must always hold.

So when you calculate the sides of these triangles in the end, you must always get sides that always obey Pythagorean theorem. Okay. Last thing I want to say is that the, or for this board anyway, is that the sine and the cosine that you get, of course, we all know it's the ratio of the triangles and all that stuff.

but it is really only a function of the angle. The sine that you get when you have a really, really shallow angle, like let's say one degree angle, if you put that in your calculator and get the cosine of it, you're going to get a number that's going to be much higher than 0.8. It's going to be much closer to one because most of the force will be in the x direction.

If I have a one degree angle like this and I take the sine of one degree, I'm going to get a number really close to zero because there's very little of that force in the vertical direction if I have an angle really small like this. In fact, let's check these ratios by calculating the sine and the cosine. Remember, the cosine is the adjacent over the hypotenuse, and the sine is the opposite over the hypotenuse.

So now that we have figured all this stuff out, let's go over and do that, and I want to do it right below here. The sine of an angle is going to be equal to the opposite over the hypotenuse. That's what the definition, the mathematical definition of the sine is.

But in this triangle, the opposite to this angle is 7.2 newtons. The hypotenuse is 12 newtons. So the sine of the angle that we get when we divide 7.2 and divide by 12, we get, what do you think, 0.6. That's what we already know the sine of it is. Okay?

And then the cosine of the angle is going to be equal to the adjacent over the hypotenuse. But the adjacent side of this triangle, adjacent to the angle, is 9.6, and then we'll be divided by 12. 9.6 divided by 12. So the cosine of this angle, when you divide those two numbers, you get 0.8. So you see everything is fine. I took the angle, I stuck it in a calculator, I hit the button, I got the sine and the cosine. I use the sine and the cosine to multiply by the hypotenuse and calculate the other sides of the triangle.

But once I have the other sides of the triangle, I can run them back through the definition of the sine and the cosine. I'm going to calculate the same numbers because everything is self-consistent like this. All right, so when I take a number and I stick it in the calculator, what the calculator is doing is it's kind of inventing a triangle in memory and it's dividing sides of a triangle to tell you what the sine and cosine are. All right, one more thing I want to talk about. Here we started with a triangle that was this big, five newtons long in the force.

Then we stretched the triangle to 12 newtons along in the force at the same exact angle. Now what I want to do is I want to close the circle all the way back and say let's shrink it down. Instead of making the triangle bigger, let's shrink it down and make it where the actual force is not 5 newtons and is not 12 newtons. Let's make the same triangle at the same angle, but with the force only being actually one tiny little newton, one little newton of force. All right, so let's shrink that triangle so that now it's in one newton of force.

So what we have there would be a triangle that would be really small. a little baby triangle like this, right? So what we would have is one newton of force acting up at some angle.

What angle would it be? Well it would be the same angle, 36.87 degrees. So if I had a little baby triangle like that with a little tiny little force but at the exact same angle, how do I figure out how much of the force is in the horizontal direction and how much is in the vertical direction?

I do the same thing. I calculate the sine and the cosine of this angle and then I multiply by the hypotenuse. So let's do it real quick.

Now, if you were to stick this in a calculator and hit the sine and cosine button, we're going to get exactly the same numbers. We already know the sine of this angle and the cosine of this angle. Specifically, we know that the sine of 36.87 degrees comes out to 0.6, and we know the cosine of 36.87 degrees comes out to 0.8. Those are the chopping factors. How much does the ratio exist in the vertical direction?

what is the ratio, how much exists in the horizontal direction. So for the y direction, what do I do? I'm just going to say that the hypotenuse is 1 times the chopping factor, exactly the same calculation we did before. Remember before it was 12 times the sine, 12 times the cosine.

Here I'm going to say 1 times the sine. So what I'm going to get is in the y direction 0.6 newtons. In the x direction, in the x direction or the x component right is going to be equal to the hypotenuse of one times the chopping factor in the x direction which is 0.8. Multiply that out what do you get x direction 0.8 newtons. Okay why do I care about this?

Because remember what I told you a long time ago. Let me pull my boards back and try to tie it all together for you. I said I was very careful I said the sine of an angle is the chopping function or the chopping factor that exists for the y direction assuming the length is equal to one.

I said that the cosine of an angle is the chopping factor or the chopping function in the x direction that chops the hypotenuse down and tells me how much I have in the x direction assuming the length of the triangle is equal to one. That's why I take the actual hypotenuse of the triangle and I multiply by the chopping factor because if this exists assuming the length is equal to one. but the hypotenuse is five times bigger than that, that chops it down and gives me the actual X value, and it chops it and gives me the value in the Y direction. Now we're going all the way to the end, and let's say let's shrink the triangle all the way down so that it really is only one newton long. If I take the sine and multiply by one, then I have 0.6 newton in the vertical direction.

If I get how much of it is in the horizontal direction, I'll take the chopping factor in the X direction, the cosine, multiply by one. and I'll know that 0.8 Newtons is in the horizontal direction. So 0.8 Newtons like this, this is 0.8 Newtons.

And over here, this is 0.6 Newtons. So you see what's going on is when I define the sine and the cosine, the sine is gonna be 0.6 divided by one, which means the sine is 0.6. The cosine is gonna be 0.8 divided by one. The cosine is 0.8.

So the cosine and the sine. really are the chopping factors assuming the length of the triangle is just equal to one. That's what they're doing.

They're saying, hey, your force is really equal to one. This is how much is in the x and in the y directions. Now, if your triangle is anything other than one, which it always is, then you just take those chopping factors and multiply by the hypotenuse, the actual hypotenuse you have. Then you get how much of it is really in the x direction. You can do the same thing in the y direction.

Take your chopping factor and multiply by the hypotenuse. to get the actual amount that's for this triangle when it has a length larger than one. Okay?

And this is why when I take the cosine and I multiply by the hypotenuse of these other triangles, I get how much is in that direction, which is the projection and so on. Because when I shrink it down like this, that's what the sides actually come out to be. So we have done a ton of So much so that I want to spend here one or two minutes just going through all of it again, because I think it really helps to see it and hear it a few times.

Let's say I'm pushing a box at some angle, a length of force of five newtons. I know that a 3-4-5 triangle is special, and it's a right triangle, the sides of a right triangle, so I label it there. The sine is defined to be opposite side from this angle divided by the hypotenuse, whereas the cosine is defined to be the adjacent side divided by the exact same hypotenuse. So in this case I get 3 over 5, the other case I get 4 over 5, and it's literally the ratio of how much is up compared to the total force, and this is the ratio of how much is horizontal compared to the total force.

A handy way to think about it is the sine of the angle is the projection to the y direction. The cosine is the projection to the x direction, so sine goes with y. Cosine always goes with x. Always. I want you to remember that.

So if we look at the sine in our case, we've got 3 fifths which comes out to a decimal of 0.6. That means that 0.6 of the total force is in the y-direction as a fraction. 0.6 of the total force.

Another way of saying that is the sine of 0.6 is called the chopping function or the chopping factor in the y-direction assuming the length is 1. So if your length really isn't 1, just take that ratio and multiply by how long your hypotenuse really is, and you will get how much of it is in the y direction, right? And same thing here. Here's your chopping factor in the x direction. If I take that number, assuming that the length is actually equal to 1, that's how much of it would be in the x direction. But my triangle is 5 times bigger than that, so I multiply by the 5, and then I get the actual amount in the x direction for my triangle.

Then we take the exact same triangle, which we now know the angle is 36.87 degrees, And we make it larger so that I'm not pushing with 5 newtons, I'm pushing with 12. And we do the exact same calculation. If I take the chopping factor, which is this, and I multiply by the hypotenuse, I get the amount of force in the y direction, 7.2 newtons. If I take the chopping factor and I multiply by the actual hypotenuse, then I get exactly how much of this force exists in the x direction.

Cosine goes with x, sine's the projection in y, cosine's the projection in x. I get these numbers, I run them through that Pythagorean theorem, and I figure out, yes, this does perform a right triangle. And then I actually go and calculate sine and cosine again using the ratios, and I find that the sine and the cosine that I get exactly match what I got from the calculator before.

And then we closed out by saying, let's shrink the triangle so that the actual hypotenuse really is only one newton. We do the exact same thing. We take the chopping factor, this, times the hypotenuse. We take the chopping factor in the x direction times the hypotenuse, And we find out that if the hypotenuse is 1, then the y direction has 0.6 newtons, and the x direction has 0.8 newtons.

So for every one unit of length of the hypotenuse, this is the amount, the cosine is the amount of that force in the x direction, and this 0.6 is the amount of the force in the y direction for every one unit of force that you have. That's why when you multiply those chopping factors times the hypotenuse, that gives you the whole projection in those directions. So, I really encourage you to watch this two times. It's a lot, and it's easy to look at it and say, oh yeah, yeah, I get it.

But what's gonna happen is we're going to introduce so many new concepts, and calculating different sides of triangles, and then you're gonna get into more advanced classes, and do things with vectors, and all this stuff, and then maybe three months from now, you might say, oh, I get it. I know why sine is like that. I know why sine goes with the y direction. I know why cosine goes with the x direction. I'm trying to bring this up to the beginning.

so you know the point of it. Because when you're solving a problem and you're trying to throw a baseball or send a probe to Jupiter or whatever, you want to take the curved trajectory, you want to split it into different directions so that you can analyze the different directions separately. So when we know that we have an angle here with some kind of number like a force, we want to split the directions to see what happens in those different directions. The rest of trigonometry and precalculus and a lot of calculus really is built on this fundamental idea. So watch this several times, go through these calculations yourself, and then follow me on to the next lesson.

We're going to get a lot more practice with the fundamental meaning of sine and cosine.

Transcript for:Understanding Sine and Cosine Concepts

Transcript for:
Understanding Sine and Cosine Concepts