 gang let's do a little bit of trigonometry okay let's do some trigonometry okay now trigonometry at the beginning stages of university at high school basically so check this out in grade eight nine even in grade 10 they tell you trigonometry is this trick and it is and it is right so great eight nine and not great great we'll talk about high school grade eight nine and ten so grade eight nine and ten grades right they say trig is about right angle triangles in canada anyway they tell you trig is about right angle triangles in grade 11 and 12 11 12 you realize trig is really about circles that's one of the important things you have to really appreciate especially when you're getting into grade 11 and grade 12 because when we're trying to study circles right let's assume you're trying to study circles and we've talked about this why would you want to try to study a circle right you would try to study a circle because a circle represents the ideal cyclic function because let's assume you stand here you're here and you're moving around if you're here you're moving around then you've gone one cycle right now the reason we want to study cyclic functions is because cycles are everywhere in our world everywhere in our world not just the earth revolving around the sun rotating on its axes the moon going around having some kind of orbit the tides of the ocean going up and down right it's not just the physical part of the cyclic nature of life it's also embedded within biology our systems are mattered that we occupy okay it's also embedded within economics huge huge you can actually invest money in cyclic companies right invest when the cycle is down cycle goes up you make you sell it you buy and there's trillions of dollars really trillions of dollars being traded on this cycle we talked about this right we did a whole thing based on personal finance right investing in personal finance we have a playlist on censor too that talks about the cyclic nature and these cycles can vary depending on if you want to look at it on the micro scale or the macro scale right are you just looking at it as a if you're a trader as a day trader or you're investing for your retirement on a long scale right are you looking at it based on millisecond trading which is there's a lot of programs out there most stocks on the market are traded based on our automatic this machine is doing it right minute 10 minute day week month year decade is that what you're investing in what cycle what speed are you investing right is it going to take you you know one second to make this cycle or is it going to take you one day to complete the cycle is it going to take you one week one month one year one decade right doesn't make a difference in regards to analyzing the circle because if you're studying a cyclic nature it applies to all of these right you don't care about the length the time it takes to complete cycle you just want to know how to analyze the cycle right apologies if i'm not reading the chat game because i want to get this train of thought out of the way right so one of the reasons we study circles is because they are the ideal cyclic function because if a cycle fits this model right or if we can create a base model right mathematical model right that we can analyze based on the ideal cyclic function we can take that and apply it to multiple systems within our society may it be based on economics politics biology right nature doesn't matter right that's the reason we study circles right so what is the one thing you do when you study cyclic functions okay what is the thing you do to use study cyclic functions you take the ideal cyclic function let's erase those you take the ideal cyclic function let's create another circle right you take the ideal cyclic function which is a circle you find its center and you put it on a grid you break it down right that way you can put numbers on on your circle right so we put on a cartesian coordinate system that's what this is called cartesian coordinate system x and y axes and we say okay if we're standing here right and if we're going to go around the circle how do we analyze that the way you analyze this the way you analyze this is let's say you want to move here you can say okay go up a certain angle a certain distance along the arc length of the angle right or you could also do this because you put on the cartesian coordinate system you could say create your right angle triangle and that links these guys up right that's how the triangle right angle triangle is connected to a circle you create a right angle triangle and this becomes your x and this becomes your y so the coordinate here is now x and y right so on a circle right if you want to know where to go if you're standing here right I could tell you to go a certain angle on the circle at a certain radius from a certain center point and you end up there or I could give you the coordinates of the circle of where you want to end up right easy right one of the things we do in mathematics or mathematicians do right they will try to simplify calculations as much as possible and the easiest number to deal with is the number one right so one thing they do they take a circle they're trying to analyze this trying to simplify things right you call this a unit circle a unit circle and what is a unit circle it's a circle where the radius is equal to one right so we're gonna take this and say the radius r is equal to one okay when you're talking about a circle that has a radius of one you call the unit circle that's it simple calculation now we want to analyze this we want to figure out what happens when you have to deal with a cyclic function with something that repeats right well you could do this you could say you know what because this is based on a Cartesian coordinate system right we're gonna have a couple things that happen one of them is our x value here is going to change as we move along the circle right because this x is basically going to move this way or this way depending on where you are here right as you move along this x is going to move along right and if the radius of this thing is one then based on a Cartesian coordinate system the point here if the radius is one you bring this thing down well the x point is one that's a given right because we're dealing with a unit circle it's got a radius of one so this link here is going to be one and the y the y value here is zero so you can say okay you know what I want to I want to find out what happens right what happens to my x value and my y value as I move around the circle right as I move around the circle I want to know what happens so for example let's take a look at this what's our x value here and why y value there you guys know and then try to figure out tell me what the x value is here and what the y value will be here and what the x value will be here and what the y value will be here what do you guys think what is it going to be right because once you see it you cannot unsee it once you see this you cannot unsee what's going to happen if the radius of this thing is one as you move this way right it's the radius exactly running right so you move along here well the x value from here it's starting at one and it's getting smaller smaller smaller smaller smaller here at zero so the x value becomes zero the y value started off here at zero and work this way up right and if the radius is one that's one negative one and zero because you're coming this way the x value of zero here it's going in a negative direction so this becomes negative one and zero right and as you move down right the y value went from one reach zero and it goes to negative one so this becomes zero and negative one so what you can do is say okay cool we got some base coordinates for this unit circle for the x and y value right so let's take this information and create another graph okay and we're going to do this yeah we'll do this here my line is not straight there let's make it straight so first of all yeah let's make two how many degrees is it all the way around the circle okay the whole imagery number derivation also comes from this imagine oh yeah yeah yeah yeah so take a look at this right 360 degrees right so if you move all the way around the circle if you turn around we just did a 360 alligah 360 ronnie pro quo everybody's 360 we know is 360 all the way around german minister doesn't german minister thought the 360 went met 180 that's the level of intelligence right we're the smart ones put on a plurus xbox 360 baby ronnie says one down few full put on a plurus right so a full circle a full cycle right and that's what we're going to call it one cycle takes 360 degrees right so we want to take a look at one cycle because as soon as we can figure out what happens in one cycle we know what happens in every cycle because it repeats brilliant brilliant right we figure that out we figure it out for a thousand cycles infinite cycles we can figure it out for the backwards going around right so let's look at one cycle one cycle is 360 degrees right so on the x-axis we're going to put theta degrees okay so let's put the numbers on there for now we go from zero to 360 degrees zero to 360 degrees and the reason i'm making two graphs is because we're going to do two graphs right we're going to graph the movement of the x coordinate what the x does and what the y does right because as you're moving along when you move here your y value is here when you go there your y value is that doing right so that's what we're going to look at let's put the y value in the top and the x value in the bottom graph so we're going to call this y and we're going to call this x one thing you want to do you want to find out the range okay per quad what trigonometry can be interesting whoa mind blown sign sinus or cosine waves just shifted so it's just shifted indeed we'll talk about it take a look so let's put the y up top x in the bottom graph okay and let's talk about the range that we're going to do right because what we did right now we said we defined the parameters right like if you're playing the game like a soccer football game basketball game but you draw the map right Robert Anton Wilson or was it Timothy Leary they called it the mind map or something like this the map of the oh i forget the terminology for uh what do you call it Robert Anton mapping something right so for the degrees we went from zero to 360 because after 360 repeats right okay what's the range that the y can go well if we start here the y can be zero right it goes all the way up here and the y reaches a maximum of one right and then starts coming down again right reaches zero again goes down again all the way to negative one and then goes up again to zero so the limit the boundary that our y value can exist in is between one and negative one so let's put our limits on that one and it can go from start from zero go all the way to one come back down to negative one go up to zero right well where does it do this where does it do this well it does it the maximum points right occurs here and that's 90 degrees right if you go from there to there that's 90 degrees the minimum point occurs at 270 degrees but one thing you should have noticed that we're talking about a circle right that's the beauty of it right once you apply it on a grid and take a look at it break it down right when you're a kid when you take things apart you can understand what they are right or try to understand what they are what their components are and then maybe you could create something new or put it back together again maybe you could put it back together better right so when we're taking a look at a circle right going around and around and around before we put a grid on it it was a whole circle we were looking at and it could have varied from one location to another but as soon as we put a grid on it we realized that hey take a look at this thing in this quadrant this quadrant right when we break it into a quarter this thing is also sort of mirrored here and then it's a mirrored here and it's mirrored there so logic says if we can understand what's happening in this quadrant we can understand what's happening here here and here cool so these four quadrants really make up the full circle when we can understand the first quadrant we can pretty much understand the second quadrant the third quadrant the fourth quadrant right our critical points really to divide these quadrants are the following this point this point this point and this point so on the theta axis here on the theta axis here let's put the degrees on here because we went from zero to 360 degrees we went all the way around so let's break this into the quadrants that we're interested in so this quadrant here this point here is at 90 degrees you come this way that's at 180 degrees you come all the way here that's 270 degrees and then if you go all the way that's 360 degrees right so let's break it down into the four quadrants that we have so if you want to break zero to 360 down to four quadrants so I put out a video on this how to break a line into pieces a long time ago like 10 years ago I put out this video how to break a line into pieces because we want to break it into four pieces is even cut it in half so we're going to cut this in half and then we're going to have two quadrants here and two quadrants here and two is even so break it in half again break it in half again and then what we got this becomes 90 180 270 and then 360 zero degrees 90 degrees 180 degrees 270 degrees 360 degrees right okay cool we're going to do the same here let's line them up 90 180 270 all right okay cool now we're ready to put our points on here because we have our this is called the domain for the x-axis but anyway we have our boundary for the x we got our boundary for the y and we got the quadrants marked off right and the x also does the same goes from one to negative one right goes all the way to one and then comes back to negative one one negative one one negative one right so let's do this we're going to break the x as well from one all the way to negative one okay now let's figure out man this is relaxing awesome relaxing math is the best math all right so what we want to do is figure out what's going on these quadrants where we are on here right here for the y we're at zero when we're at zero degrees the y is zero so we're here actually let's do this in different color let's do this in green we like green let's do green who knew learning math was so relaxing no wonder all the students fall asleep in math I some I hope you're not finding this boring though this is relaxing but not boring I hope right unfortunately sometimes you don't get that that aspect of it when you're sitting in a classroom air conditioning going buzzing 30 people some don't care some of the people sometimes that don't care are the instructors and usually the curriculum is set up not to be exciting Ronnie I love my me too so when we're standing here right our y value is zero cool we're gonna put it there let's look at the nodes here when we're standing here our y value is one so at 90 degrees we're at one let's go to this node at this point our y value is zero and we're at 180 degrees right so 180 degrees we're back down to zero over here we're at 270 degrees and our y value is negative one and if we go back to zero again our y value is zero and we're at 360 degrees right so we're back here now take a look at this thing you might look at this and go oh so the graph must look like this lines going straight but it doesn't one of the reasons it doesn't is because this is curved so the way we connect these dots is not just lines going like this it's curved now if you love music that should be you should know what that is and that's sound wave really if you like going to the ocean swimming you should be familiar with this that's waves in the ocean if you're trading stocks you should be familiar with this type of motion that's trading eyes and lows right if you understand what light is you'll know that light is a wave particle wave right oh my god we do this math in my wing chump train really very cool very cool green for hope awesome for qua not green for a stupid climate one no can't wait to see chucho draw the perfect waves awesome we make a chucho waves ah nice nice nice sweet sweet sir wise brought salutations right so this this this wave has a name okay we call it a sine wave let's do it for the x as well right where we are on the x-axis as we move around the circle what our x position is hard to do it with two things how do we do this we go as we're moving around well you can't even see my other pen as we're moving around oh my god so difficult to do is like an amusement park thing when you're doing your thing right the x-axis looks like this when we're at zero degrees we're at one when we're at 90 degrees we're at zero when we're at 180 degrees the x is negative one when we're at 270 degrees the x is zero again and we're at 360 degrees we're at one again so this wave for the x-axis looks like this now remember this thing keeps on going around and around and around so this doesn't end here this continues like this and does this it's basically picking up from here and going like this so this part goes like this over here it's picking up just like this and going like this again right so it continues same with this right and it comes down this one is called a cosine wave cosine wave sine wave now how is this related to triangles well it kicks into so katoa sine cosine and tangent and stuff so if you study triangles right in grade eight nine and ten you learn about opposite hypotenuse and adjacent right so if you have let's do this in red if you have a triangle and here's your angle you call this side the opposite you call this side the adjacent to theta and you call this side the hypotenuse okay luni woo thank you very much for the follow right so you learn you learn that sine theta sine of this angle is the opposite side divided by the hypotenuse opposite divided by hypotenuse okay now keep this in mind this is this is a ratio right so what it's saying is this it's saying this side divided by this side is sine theta right so sine of this angle is defined as the ratio of the opposite side divided by the hypotenuse okay why is this important because no matter how big your triangle is or how small your triangle is so let's draw another right angle triangle so we have two triangles here right we got this and we got the bigger triangle and this side is again the opposite side from this angle and this whole length again is the hypotenuse goes all the way to there right and this one is the smaller number so sine of this angle it's still going to be the opposite side relative to the hypotenuse so we don't even need to draw the hypotenuse twice because it's the same hypotenuse so I'm going to erase this and just call this height and the hypotenuse depends on which triangle you're talking about right so sine of an angle is basically saying that hey the ratio of the opposite side to the hypotenuse is going to be the same no matter how big or how small the triangle is now we've got three triangles that's the opposite that's the hypotenuse this ratio is going to be the same so for a given angle for a given angle let's say let's say we have 30 degrees right yeah let me write that bigger so you see it better let's say you have 30 degrees let's say you have 30 degrees right this side divided by this side would be the same as if you had a smaller triangle and this was 30 degrees we'd be the same as this side divided by this side okay so if this was the same as this and this was a and this was b right and this was x and this was y the sine trigonometry tells us this that a over b a over b would have to be equal to x over y oops x over x over y not y over x x over y x y is this important wow this is they're called similar triangles this is those models when you're building models and stuff if you're into collecting anything or if you're into engineering interested in engineering or models or anything really right when you look at it when they say oh this model is you know one to ten or ten to one right that's this is what they're talking about so for example if this was you right this is called proportionality by the way right similar they call it similar triangles in trigonometry but you could also call it proportional they're proportional right so if you have you you here and there's a little mini you right and your height is let's say six feet and your arm is two feet i don't know if that would be a legit or not two feet three feet two feet it's pretty small that's like t-rex level arms isn't it so let's say three feet right so let's say your arms are three feet and if you want to make a little mini version of you that is two feet tall then you can figure out how big the arm needs to be right because all you do you say this divided by this has to equal that divided by that so all you would do is say six divided by three is equal to two over x cross multiply you get six x is equal to six and divide by six so x is equal to one so you have to make this one your arm one feet right 10 minute warning are we into two hours already elder god wow time flies what yeah yeah we're gonna go over time a little bit a little bit sorry gang right and he should put me in detention in D3 format Ronnie says he's a good man thanks he does so i just want to make that clear that sometimes it's not clear when they teach soco to soco to you guys initially when you're studying detention detention for us so when they teach you trigonometry in grade eight nine and ten it's not clear that why is sign important is because the sign of an angle is the same value for any size triangle wow incredible the ratio of one side divided by the other side for any given angle is the same cool this also applies to costata which is adjacent over hypotenuse adjacent over hypotenuse and it also applies to tan theta which is opposite over adjacent opposite over adjacent right now take this information that you know take this information that we just learned that for a given angle the ratio of one side to another side is the same right take this triangle apply it here to a unit circle right to a unit circle and realize that a unit circle is a circle defined as having a radius one right and realize that hey wait a second this thing tells us it doesn't make a difference how big the triangle is you could have a bigger triangle and if you drew a circle we can't even do it my triangle is so big right you could draw a circle come all the way down you could take a bigger circle my circle off you could take a bigger circle right you could take a bigger circle and this graph is going to look almost identical the only thing that's going to change is the radius this is just going to change right so if the radius of this is two right we take a unit circle we multiply the radius by two right double the radius all that's going to happen this wave is going to look the same it's going to have the same motion the only thing that happens is it goes from two all the way down to negative two so it just amplifies it right it just makes it look like that wow cool now we can take circles and if we're making music you can take this music right you can take these waves and amplify them make sound cool but wait a second it's going to have the same pattern all of this is going to amplify it right well we can take the same graph right and graph it based on how fast it goes around right if remember we talked about if it takes one second one day one week one month one year one decade if this zero to three sixteen instead of being an angle if it was time how long does it take for it to go around right what you can do is create waves that are not only amplified differently bigger or smaller but also change in frequency the period changes this is called a period how long it takes to do one cycle right so all the sun you could do obviously it should be a better graph should be right maybe we could do it here faster cycle right and what you can do with these things you can multiply them you can add them you can move them you can translate them right you could do a lot of things once you understand the base mathematics of trigonometry these functions these formulas what you can do is you can take what color is going to stand up you can take a general sine function you can say it's a function right here let's do a little bit of erasing we'll kill this right so our function up here is f of x is equal to sine theta right that's what it is f of x being your y right what you can do is manipulate this thing you could go f of x is equal to a number so we know what this looks like this guy looks like our green graph right well what if we want to graph the following function negative 2 sine i talked about pi yet sine um three theta plus four minus actually plus six right so let's say this is our function we've taken our original function and we're going to do this to it well what does that do well the negative 2 flips the original function this way and it makes the radius 2 so it amplifies it by a factor of 2 this guy three here compresses the function to a third of its period this guy here moves at four units back right this guy here moves at six units up so we're taking our original function and manipulating it and this can be done with to model certain either physical or physical systems like a ferrous wheel right you can model a ferrous wheel based on a sine function tides of the ocean you can model this way or you could use these types of things to model market reactions and stuff like this that's sort of a good intro to the power of trigonometry and why it is you study right angle triangles because you want to study circles why do you want to study circles because you want to know what cyclic functions how cyclic functions behave why do you want to study how cyclic functions behave because cyclic functions are embedded in our societies are embedded in our lives are embedded in nature they're part of life right and once you can model life man you can do a lot of things with it right you could do a lot of things okay that's trigonometry and it just explodes from there right and we have a playlist trigonometry playlist here i'll link it up for you guys on our sensor tube channel if you go to if you go to our sensor tube playlist or sensor tube channel and go to playlist i'm just going to do this math where's my trigonometry playlist uh hey trig trig oh i should put math on there too so people can find it silly me math didn't bring it up here's our trigonometry playlist see that's excellent i like how you describe the function here yeah it's super cool it is crazy cool and we we can get on it next time as well just talk about it we sort of did not a quick ending sort of brought it all together again sort of did a little summary but that's a good little intro for it and we didn't go too much over time that's good that's good and we can definitely explore this more in the future and look at different functions and actually graph something like this right and do the translation so maybe in the next math stream we'll pick up from here and just graph something like this based on what we created here we'll create a table and i'll show you how it's done super cool super cool way of doing it and so easy so easy now i'll guess she's your back see that's that's yeah and via and via they might be passed out now they might be passed out they're running around playing around and stuff like this um gang let's call this stream let's call this stream that was fun that was definitely fun your fist thank you very much for the follow salutations salutations and welcome to our live stream channel and gang do not forget do not forget free assange free assange free assange julian assange is a publisher and journalist that is being crucified for trying to bring transparency and accountability of capital's power to humanity something that we desperately need in our societies right for more information see wikilees.org defend dot wikilees.org or our junior sergeant wikilees playlist on censor tube see that thank you for the compliment see that yay if you want to know what this work is about i am on patreon patreon.com over slash gcho chy ch over you can follow the work there we have a sub stack page as well and a subscribe star page my pleasure parkour my pleasure parkour you can definitely follow the work there for those of you that are supporting this work on patreon and on sub stack for now thank you very much for the support as well as the people that are supporting us on twitch it is in large part because of the sport that we're getting on these two platforms including the handful of people that are supporting us on censor tube uh that we're able to do what it is that we are doing and i thank you very much as well as the support we're getting from the mods both on twitch on our video sharing platforms and on our little server that we built a little community that uh we're sharing information and discussing things and talking about things and trying to figure things out including talking about mathematics uh and gang we do announce these last streams 30 minutes before we go live on twitter mines vk gap parlor and getter you can follow the work there and we do have a soundcloud page where we upload certain podcasts uh certain streams podcast on soundcloud and those podcasts are available uh should be available on your favorite podcasting platform thanks was fun yeah super fun i love explaining trade so much fun it takes a fair amount of concentration to be able to link everything up and sometimes and it's different every time almost different every time because sometimes i put in more things sometimes i put in things i shouldn't have put in because i was trying to go in this direction i go on the tangent i go to come back track it's super fun it's super fun as a as an educator someone that teaches it uh has been doing it for 20 plus years i still love it because it keeps me sharp that's what mathematics does gang really you want to become smarter learn mathematics you want to become master of your own domain your own life learn mathematics it it's my it's my important recommendation to everyone right you want to be your be a free thinking human being in control of your own destiny learn mathematics aside from that gang uh i hope you have a fantastic next couple of days we're mapping global conflicts on tuesday starting at 1 pm part four join us if you can bye everyone