05 - Sine and Cosine - Definition & Meaning - Part 1 - What is Sin(x) & Cos(x) ?

well 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 precalculus 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 it's 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 of 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 going to spend some time i'm going gonna 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 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 it's 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 of 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 5 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 you know pushing in that direction what's the size of this force we're going to call it 5 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 newton it's 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 right that's i'm just breaking it up into two different directions all right so you can think of this uh 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 got to 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 uh in the past we learned some of these special triangles so i'm going to put a little divider bar here there's something called a 3 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 if i prove in 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 3 newtons you see the 5 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 is then 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 5 newton force is acting at some angle up let's say it wasn't 5 newtons let's say it was 7 newtons or 10 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 punch line 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 any other kind of uh 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 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 in 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 uh 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 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 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 a 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 five 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 sign 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 if 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 five so it's three-fifths this is a number that i can put in my calculator right and what this is telling me is that the sign 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 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 uh much total force so it's a ratio it's like you know if i told you in general if i said hey um 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 and you 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 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 uh or i should say yeah in a projection amino 2 the y direction okay so i'm going to kind of circle this because it's really really important and then the cosine is the uh cosine of theta is basically the projection projection yeah like this uh to the x direction right 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 that's the punch line 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 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 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 sign of this angle is three-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 three-fifths we just calculated that if i grab a calculator and take three fifths i get an answer of 0.6 right what this means fundamentally actually i think i'm going to do it right to the side so i can save some space here okay 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 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 uh but it's really kind of almost in the middle about point six 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 did we get over here right for the cosine of this angle what did we calculate four fifths when we grab four and divided by five in the calculator we get zero point eight 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 a number in your 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 okay but 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 1. the maximum this could possibly be is equal to 1. when i look at this i take 3 divided by 5. again 5 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 want to 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 eighty percent multiplied by a hundred what percent is exist in the horizontal uh 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 punch lines i want to pull out of this i tell students this all the time and i really want you to remember 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 book 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 a function for the y direction uh if the length is equal to one 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 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 going to 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 sign 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 assigned 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 point six and so then i get three newtons so you see the 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 just took a light and i shine it down what's gonna 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 5 unit long ruler i'm going to get a length on this side of 3 newtons so there's different ways to think about it and i'm talking about a different ways at once so that you can internalize what they mean the sign 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 of the y uh of this force exist in the y direction and also it's called a chopping function because point six 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 it's chopping the thing down in the y direction if the length is equal to one what it means is for every one newton of force that i have here point six 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 okay but if the force over here wore 1 newton instead of 5 and i 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 1 unit long 0.8 of the force would exist in the horizontal direction and if the force were really one newton long point six would exist in the vertical direction so here i'm about to get to the punch line 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 for the x direction is called the cosine it came out to point to be 0.8 in this case the chopping function for the sine is called the call it's 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 punch line 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 i'm chopping if i can spell chop okay i'm chopping the five 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 uh 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 i'm then multiply by 0.8 what am i going to get grab a calculator and do this what you're going to get is 4 newtons so the 3 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 then point eight of it would exist in the uh horizontal direction right but the length is not one newton the length is five noon 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 okay it's assuming that the the length of the force is actually one but the length isn't one it's five times bigger than that so i take the chopping function and 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 sign what is the sign if i go back to my definition the sign was opposite over hypotenuse then i multiply by the opposite over the hypotenuse that is what the sign 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 sign means the hypotenuses cancel and i get back the opposite side which is in the y direction if i look at this and 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 hypotenuses 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 cancel 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 hypotenuses cancel 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 i'm not going to actually tell you why i know because it becomes 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 30 6.87 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 sign 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 that 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 uh shape 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 fourths so originally i had the box over here and i was pushing at an angle of 36.87 degrees and 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 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 0.87 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 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 one newton it's the fraction it's the amount of the force that would be in the vertical direction if i only push with one newton but i'm not pushing with one 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 okay 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 one but it's not one 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 too 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 what 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 0.16 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 they 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 ends 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 sign that you get when you when you have a really really uh 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 1 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 sign 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 sign 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 divide 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 the 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 stretch 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 it's 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 four 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 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 okay specifically we know that the sine of 36.8 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 is 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 sign 12 times the cosine here i'm going to say 2 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 1 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 sign 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 the actual hypotenuse of the triangle and i multiply by the chopping factor because if i 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 of 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 1 newton long if i take the sine and multiply by 1 then i have 0.6 newton in the vertical direction if i get the 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 point eight newtons is in the hor 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 going to be 0.6 divided by 1 which means the sine is 0.6 the cosine is going to be 0.8 divided by 1. the cosine's 0.8 so the cosine and the sine really are the chopping factors assuming the length of the triangle is just equal to 1. that's what they're doing they're saying hey your force is really equal to 1. this is how much is in the x and then the y directions now if your triangle is anything other than 1 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 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 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 material 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 a force of 5 newtons i know that a 3 4 5 triangle is special and it's a right triangle the sides of a right triangle i label it there the sine is defined to be opposite side from this angle divide 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 sign 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 sign in our case we got three-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 one 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 five times bigger than that so i multiply by the five 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 exact exactly how much of this force exists in the x direction cosine goes with x sine's the projection and y cosines of projection and 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 law 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 is 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 uh this .6 is the amount of the force in the y direction for every unit 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 and say oh yeah yeah i get it but what's going to happen is we're going to introduce so many new concepts and calculating different sides of triangles and then you're going to get into more advanced classes and do things with vectors and all this stuff and then maybe you know 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 like throw a baseball or send a probe to jupiter or whatever you want to take the curve 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

Channel: Math and Science

Views: 906,303

Rating: 4.8794661 out of 5

Keywords: sine, cosine, sin(x), cos(x), sin, cos, tangent, tan(x), tan, what is sine, what is cosine, cosine definition, sine definition, sin and cos, sine meaning, cosine meaning, trigonometric functions, circular functions, trigonometry, trig, calculus, algebra, math, cosine law, cosine rule, right triangle trigonometry, pythagorean theorem, sine function, cosine function, sine graph, cosine graph, sin graph, cos graph, trig functions unit circle, trig functions identities, unit circle, tutorial

Id: vuoNyvMvDtA

Length: 48min 5sec (2885 seconds)

Published: Tue Jun 23 2020