 So we've reached an important point in our discussion of what trigonometry is and how we're going to analyze Triangles and circles and talk about cyclic functions and take a look at what waves look like on a graph, right? And you know, we've done a fair bit so far and we'll do a little recap We'll do a little setup, but for anyone that wants to skip the setup and you've already been following the videos Just to let you know what we're about to do I'm gonna draw another unit circle here. Okay, and We're gonna create a table here and the table that we're going to create is What the three primary trig? Ratios are for the unit circle as we move around the unit circle Okay, and we're gonna create the table for the special right triangles and The four quadrants the nodes, right? For theta and radians, right? So we're gonna find out what the three primary trig ratios are for 030 1690 and multiples of that all the way up to 360 and the equivalent of it in radians, okay, and this table that we're going to create is Something that I tell all of my students that as far as I'm concerned every trigonometry test in the world the first question asked should be Generate this table and this isn't about Memorizing this table a few because if you're trying to memorize mathematics, you're gonna fail Mathematics is not about memorization. It's about understanding. It's about using right? You don't memorize a language. You learn how to use a language mathematics is the same thing especially when it comes to This table that we're about to generate, right? So this isn't about memorizing this table This is about learning how to generate this table and I'm gonna take, you know Fairly long time explaining how we get all the answers get all the ratios or the three primary trig ratios for the unit circle in This video and then what I'm gonna do I'm gonna produce there's two more videos coming of the same thing and one of them is gonna be a little bit longer than the other one and yet The shortest one is just basically me without giving any commentary and just generating the table and what you should do is If if you know how to generate this table, then fantastic we can continue on with What you know our discussion of trigonometry if you do not generate this table You need to take the time to learn this table and this table basically talks about You know what our ratios are for a unit circle what the three trade ratios are how we convert between radians and degrees What the value the x-coordinate and the y-coordinate are for us moving around the unit circle and these are all really sort of the basis of what we need to know before we can actually start using trigonometry, right? start looking at certain systems, okay, and I'm making these three videos basically because I want there to be a you know extended discussion of How to generate this table and then there had there needs to be a video of walking through how you can do it rapidly and One as a sort of a quick review for anyone about to writing the test of how they go about doing it, right? Because what I what I do actually tell my students is whenever they're writing trigonometry exams for grade 12 specifically higher level trig, right? When they get their tests for them just to put their name on the test, right? so if you're writing a trig exam as soon as you get the test you put your name on the test and you put that test aside and The first thing you do is you take one of the scrap pieces of paper available to you and you generate what we're about to generate Okay, because that information is going to Prevent you from making any silly mistakes during the test And it's also going to provide a lot of the answers to a lot of questions in most trigonometry tests, right? Because if you know how to generate this table, you understand the three trig ratios You understand the unit circle you understand degrees you understand radiance how to switch between the two You understand what the trig ratios mean, which are basically your x and y coordinates, right? So basically by learning how to generate this table you Understand almost everything we've talked about in the previous videos and now sort of a critical point where we can progress further, right? Because as soon as we understand this as soon as we know how to generate this there's two different avenues We can go and trigonometry one of them is sort of We did a little teaser. Well a long teaser when we graph the trig Functions right when we graph the sine cosine and tangent functions So one thing we're going to do when we finish, you know Learning how to generate this table is go down the avenue of taking a look at trig functions And how they you know how they model certain systems and start graphing certain system, right? The other branch you can go down and trigonometry is taking a look at trig identities And we will do that as well, right? So as soon as you learn this How to generate this table there's two two different avenues you can go in your study of trigonometry And it's up to you which direction you want to go For me, I'm going to take a look at the functions first and then we're going to look at the trig identities because Trig identities is more playing with the language and the relationship between the different ratios in the unit circle The trick functions take looking at the trick functions and how to graph them and what they represent is more taking trigonometry Applying it in the real world and taking a look at certain systems that want to analyze, okay? So that's what we're going to do for anyone that wants to skip the setup for now What we're going to do is I'm going to create another unit circle here I'm going to set up my table here and there's going to be you know a little reminder here that At the information we're going to present here in shorthand and short form for us to generate this table And what we're going to put on this page is what you need to be able to generate Before you write any major trigonometry test, okay? So first thing we're going to do is Create another circle, right? So what I'm going to do is Go Find my center here again, and I got my Floss, where's the loop? We got the loop here. I think the loop is getting pretty tight there And I'm going to throw this guy on here and then we're going to go around, right? So let's go down again. I think this was 24 done enough. I should know what it is, right? So 1 2 3 4 5 6 7 8 9 10 11. So here's 12 and then we're going to go 1 2 3 4 5 6 7 So what we're doing now is a little bit better This other pen, so an off-circle a little bit, but that's okay Can't make it all perfect, right? So Now that we've got the circle set up, let's go Set up our grid. Yeah Because that's what we want And we're going to set that up Using our little lever again, right? Green axis here X axis here So what we're going to do we're going to talk about theta and degrees radians sine cos and tan and these are going to be our rows, right? And what we're going to do I'm going to create the table, so I'm going to leave a little space So let's do this We got this, we got so what I've done so far is Generate it our unit circle, the circle put it on a Cartesian coordinate system, right? And what I've done is created a table and usually This table I generate, you know, it's one long table, right for going from zero degrees all the way to 360 degrees But because we're limited on the space that we have here I've broken this down into two segments, right? And what we're going to do we've broken this thing down into the rows being radians degrees sine cos and tangent, right? our three primary trig ratios angle and degrees and angle and radians and What we're going to do we're going to walk around The unit circle so we're going to start off here, right? As we have in the past videos, right? And we're gonna move around The circle, right? And we're going to look at what happens to our three primary trig ratios As we move around the circle, right? We're going to look at the ratios and the ratios as we've talked about before as we move around this, right? If this is one two three four five six seven eight nine As we move around let's say we go here Right we end up here The three the two primary trig ratios anyway the sine and cosine are just basically our y and x-axis, right? Our x-axis being cos theta our y-axis being sine theta, right? As we've talked about before So what we're going to do we're going to take a look at our coordinates, right? As we move around the circle, which basically means we're going to take a look at what our y-value is and and what our x-values are going to be, right? At each of the stops for the special right triangles and the nodes, right? And it's gonna be you know, we're gonna record it all figure it out Throw it on here, right? And what we're going to need to do is know how to generate this before we continue on in studying trigonometry, right? Because once you know how to generate this it means you understand everything else we've talked about previously and that's the key, right? We just built the foundation of trigonometry, right? And what we're doing right now is Saying that we know how to you know what it's all about What the what the trigonometry what trigonometry the system is, right? We're we're stating that we understand how triangles are related to circles We're stating that we understand what it means to move around the circle, right? How our coordinate changes We by generating this we state that We understand what a cyclic function is and how wave graphs, right? And we're going to do it again for radians because we've already sort of jumped the gun and graph sine cosine and tangent for theta and degrees where we're going to do the same thing again after we generate this table For radians, right because we're going to mainly work with radians So I can't emphasize this enough For us to progress in studying trigonometry either going down The the path where we're going to look at functions graph functions and take a look at how cyclic functions You know, we can manipulate cyclic functions or how cyclic functions Are related to you know the unit circle trigonometry, right? or if we're going to go down the road of Trig identities, right? We're messing around with the system and seeing how things are related, right? No matter which branch you're going to go off You need to be able to generate this table Okay, and as I stated before as far as I'm concerned the first question on every trigonometry test should be Generate this table not memorize but generate because if you're memorizing this it doesn't mean it means nothing really. It's You're not saying that you understand the system You're just saying that you know how to memorize something now the question is do you know how to use this, right? And the reason the first question in every trig test should be generate this table is because We're going to use this table for almost everything we do from now on Right, so instead of trying to always do single calculations Right figure out what a certain ratio is what we're going to do is we're going to have this table as a reference, right? So when given a certain the radian we can find out what it is in degrees We can find out what the trig ratios are right when given a certain trig ratio We can go up and find out how that stuff is related to the other trig ratios as well as the angle and Degrees and radians right so it's going to be really handy and most trig tests that you end up writing a Lot of well many of the questions Are just basically this table parts of this table be an answer from this table, right? Then this is basically for our special triangles as the nose So what we're going to do is we're going to Lay down the degrees first because that's the one we're really familiar with right and As we talked about the first degree. We're going to look at is Zero and then we're going to go to 30 degrees 45 degrees 60 degrees and 90 degrees, right? And that's the first quadrant Right, so we're going to go with theta in degrees is going to be 030 45 60 and 90 right and One of the most beautiful things about trigonometry is not that well for sure that it it's Circles it represents the ideal cyclic function that once we start here We go all the way around We do it again, right? So as long as we understand one rotation we understand every other rotation, right? So Circles unit circle is a cyclic function it repeats. So there's huge symmetry within this, right? one of the other beautiful aspects of a circle is is if we understand what's going on in this quadrant then we understand what's going on in The other three quadrants with slight modifications, right? Because this thing This quadrant here we can just mirror it here and then flip it down and then flip it down And that's what's gonna happen for this table all we really need to learn is what happens in the first first quadrant because everything else is Sort of copied from this is generated from this With it either being positive or negative right or changing the sign being negative because all of the stuff in this quadrant is positive, right? so What we're going to do is Learn what's gonna happen in the first quadrant and then take a look at what happens in the second third and fourth And you'll see the beauty of it how symmetrical it is, right? So the first angles we had we had zero degrees 30 degrees 45 16 90 The next special angle that we have on our unit circle is we're gonna go 90 and we're gonna add 30 Right, and then we're gonna add 45 and then we're gonna add 60 and then we're gonna add another. 90 right? And then we'll get to 180 we're gonna add 30 we're gonna add 45 We're gonna ask 60 and I'm gonna have 90 when we get to 270 we're gonna add 30 We're gonna at 45 we're gonna ask 60 and we're gonna 90 or back here, right? So that's how we're going to generate the rest of the angles in degrees, and they're just basically multiples of the first quadrant, right? So from 90 degrees, right, from here, we're going to go 30. So 90, we're going to add 30 degrees. So that's 120 degrees, right? And whenever I'm generating this table, whenever you generate this table, you're going to need both hands, okay? Or you're going to have to be able to, you know, have a good track with your eye, because all that's going to happen is we're going 90 plus 30 is the next special angle. And then 90 plus 45 is 135, right? And then 90 plus 60 is 150, and then 90 plus 90 is 180 degrees, right? That's how we generate the rest of the angles and degrees, okay? And we're going to do the same thing in these quadrants, right? Because we're 180 degrees, and then we're going to add 30, which is 210, right? Now, again, I usually generate this table when I'm working with it on one piece of paper from, you know, zero degrees to 360 or zero degrees to 360 degrees, all in one shot, right? That way I can scan it and go down the table and pick things off, right? Because we're limited on space, we're going to have to have, you know, break it up into two pieces. And by doing that, what I have to do is make this column the 180 column as well, okay? So keep this in mind. This guy, when you're generating in one sheet, and we will do it in one column or one table, this guy is just an overlap of here, okay? So this guy is going to be 180 degrees, right? And then we're going to add 30. So 180 plus 30 is 210. 180 plus 45 is going to be 225. 180 plus 60 is 240 degrees. 180 plus 90 is 270 degrees, right? And then again, we're going to add 30, 45, 60. So 270 plus 30 is 300. 270 plus 45 is 315. 270 plus 60 is 330. 270 plus 90 is 360. And that's how you generate the degrees column as we move around this thing, right? And if we're going to, you know, I'm not going to put on the lines here because it's going to get really busy, right? But what we're going to do, we're going to put the ticks for the x's and y's, right? So basically what we've done right now is let's just draw the little dots of where we are, right? So this one is going to be 30 degrees if we do one triangle here. And then we've got 45 here. And then 60 is going to be about the same distance here. So let's do here, here. Yeah, so it should be one, two, three, four, a bit, right here, okay? And let's generate the trig ratios for these guys. And then we'll continue the dots around and you'll see how that stuff lines up for the x's and y's, right? So what we've got right now is we need to figure out what our trig ratios are for these guys, right? Now one thing we're going to have to do to give us a reminder because I don't like memorizing, go into memory every time and picking out what the special triangles are, what the ratios of the special triangles are, what their sides are, what the unit circle is because I'm not going to draw a unit circle like this exactly every time I'm going to work with the trigonometry, right? Whenever I'm going to generate this table, I'm just going to make it shorthand, right? Really fast. So what we're going to do up top here is we're going to throw just a circle here, very simple. But it's going to remind us of what we're really working with, right? It's a good trigger for your memory to kick you into trigonometry mode, okay? So what we're going to do, we're going to draw a unit circle here. One, two, three, four, five, center. And I'm going to do this. I'm going to try to generate this as clean as possible. So let's go here, here. So let's throw a unit circle on here. Let's see. Yes, use this guy. It's hard to do. I tried it once before doing it with the floss, with doing a little circle. Very difficult, very difficult, right? So I'm just going to do this. Hopefully this comes out cleaner, right? So we have a unit circle here, okay? And this point here is going to be one and zero, right? This point here is going to be zero and one. This point here is going to be negative one and zero. This point here is going to be zero and negative one, right? As we've talked about before, if this is a unit circle, the radius is one. And those are the coordinates for your x and your y-axis. And your x-axis happens to be cos theta. And your y-axis happens to be sine theta, right? And if you need a reminder, you can put that down. You can say this is cos theta and that's sine theta, right? The other thing we need, we need our two special triangles that cover 30s, 45s, and 60 degrees, right? So I'm going to plop this down here as well. Let's do... So these are our two special triangles. And, you know, for lack of a better word, this is something you're going to have to memorize. But it should be something that you've used a lot. So it should be, you know, more instinct of generating this than anything, right? So the special triangles should be 30, 60, and 90 degrees, right? And across from the smallest angle, we put one. Across from 60 is going to be square root of three and across from 90 is going to be two, right? And 45, 45, 90 degrees. 45, 45, 90 degrees is going to be one, one square root of two, okay? And we talked about how you can generate these things, right? So it's not really that you're memorizing it, it's that you understand that you've worked with it enough that you know it. So what we're going to do now is take a look at what sine of zero degrees is, 35, 45, 60 degrees and 90 degrees is. And then we're going to generate the radius for this and the other two trig ratios cos and tan for this, right? So let's do sine theta first, okay? Sine of an angle is opposite over hypotenuse, right? So let's just draw one triangle here. So every time we refer to it, hopefully you're familiar, you're comfortable with it. And we've got something that we can point to, right? So we're going to take a look at sine theta first. And this is our angle here, right? And standard position, right? All the terminology we've learned. So sine of zero degrees, sine is our Y, right? So if we're at zero degrees, this guy is flat, right? The terminal arm is right there. So sine of this is what the Y value is when you're standing here. And when you're standing here, your Y value is zero, right? Sine of zero degrees happens to be zero, right? Sine of 30 degrees, right? What we're going to do, we're going to look at our special triangle, right? So for the nodes, for zero degrees, 90 degrees, 180 degrees, 270 degrees, and 360, we're going to look at the unit circle, right? For the other angles and multiples of them, 30, 45, 60, we're going to look at the special triangles. So sine of zero degrees, we're here. The Y for this at this position is zero, right? So sine of zero is zero. Sine of 30 degrees is opposite over hypotenuse, right? Y over R, right? So it's one over two, so it's a half. So sine of 30 degrees is one over two. Sine of 45 degrees is one over root two. And if you want to rationalize the denominator that we've talked about, it's root two over two. So I'm going to write down one over root two because I'm comfortable with working it that way. When we maybe we'll switch it up per 45, right? So this is going to be one over root two. Sine of 60 degrees is root three over two. And sine of 90 degrees, we're not using the special triangles anymore. We're going to use a unit circle. So we're up here, right? So 90 degrees, we're up here. And our Y value for that for a unit circle is one, right? So sine of 90 degrees is one. As for what these numbers mean, a lot of people I found they have a, it's sort of difficult at first trying to appreciate what these values mean. But if we're talking a unit circle, right? We're talking a unit circle, our radius happens to be one, right? So this point here is one and zero, right, as we laid out here. And as we're moving up, our Y value is increasing, right? Maximum point we can go to is one, right? So this is zero and one, right? We're standing. So what the trig ratios are is where we are on the Y axis. So when we're at 30 degrees, the Y coordinate is one over two. So here is a half, right? That means this point here is a half. So let's take a look at what our cost value is, right? How far along we are on the X axis, right? Costs of zero, we're here, right? We're here. Our X value is one, right? Our X is one, so cost of zero is one. Costs of 30 degrees, we have the special triangles, adjacent over hypotenuse, root three over two. Costs of 45 degrees is adjacent over hypotenuse, one over root two. Oh, sorry, yeah, one over root two. Costs of 60 degrees is one over two. And cost of 90 degrees, 90 degrees we're up here. Our X value has gone down to zero, right? And then we're going to go into the negative part, but we've gone to zero. Our X value is zero. That means cost of 90 degrees is zero. Now, one of the first symmetries you should be noticing right now is that costs and sign are mirrors of each other, right? Because if you take a unit circle, if you take a circle, really, and this is our Y axis as our X axis, if you rotate this 90 degrees, the X axis becomes the Y axis and the Y axis becomes the X axis. So it's sort of a mirror of each other, or a rotation, right? So one symmetry that we have when we're generating this table is when you generate the sign values, all you have to remember is the cost mirrors the sign. With the center point being the axis of symmetry, really, the mirror, not the axis of symmetry, but the mirror point, well, axis of symmetry as well, but the mirror point being 45. So when sign is one here, cost is one here, root three over two, root three over two, one over root two, one over root two, because that's our mirror point, one over two, one over two is zero, zero. So they're sort of, you know, mirror of each other, so as long as you can generate sign, you can generate cost, if you don't want to look at these guys, right? Now, 10 is sign divided by cost, and it also happens to be opposite over adjacent, right? So we can generate the 10 ratios from either taking sign and dividing it by cost, right? Or looking at the special right triangles. I use both for this, because for the nodes, for 0, 90, 180, and 270, I use the division, sign divided by cost. For the other angles, 30, 45, and 60, I use the special right triangles. I use these two guys. So 10 of zero degrees is sign divided by cost when we're here, zero divided by one is zero. 10 of 30 degrees is, I go to the 30 degree special right triangle. 10 is one over root three. 10 of 45 degrees is one over one, opposite over adjacent, right? So it's one, one divided by one is one, right? 10 of 60 degrees is root three over one. So it's root three. 10 of 90 degrees, I'm gonna note, I'm at the top of the quadrant here. So I'm gonna use sign divided by cost and one divided by zero is undefined. We can't divide by zero, right? One of the limitations we have in mathematics, right? So one divided by zero is undefined. So we just figured out what our three trig ratios are. As far as what the cost numbers mean, well, they're the same as the sign values, right? Oh, we forgot to put these sticks in, right? So we'll do this right now. So for the 30 degrees, so for the 30 degree angle, when we're at 30 degrees, our Y value is one over two, right? Sine theta is one over two. Cost theta is root three over two. And that's just the number, right? That's the square root of three over two, right? So if we grab a calculator, what does that mean? Square root of three over two. Let's bring out the trusty solar calculator. This thing keeps on ticking. So three square roots, where's my square root? Square root of three divided by, so square root of three is 1.73, right? 1.73. So that's what the square root of three is. Divide that by two, divide that by two, is 0.8660 dot, dot, dot repeating, right? So that's just that number is a coordinate system, right? So when I write down our coordinate here is square root of three over two and one over two, right? All I'm saying is on a Cartesian coordinate system, when you're moving around a unit circle, a circle with radius one, if you're 30 degrees up, right? Where you are is your x value is 0.866 dot, dot, dot and your y value high up you are on the y-axis is 0.5, right? That's all it means, okay? And for the 45 degrees when you're standing here, and this is our 60 degrees, when you're standing at 45 degrees, sine of 45 is one over root two, right? So your, this value here is one over square root of two and the cos of 45 is one over square root of two, right? So cos is one over square root of two, one over square root of two and that's, I think it's 707, let's do this. So one divided by two sine of function square root is 7.071 dot, dot, dot. So your coordinate here is your x and y are, your x is one over root two and your y is one over square root of two, right? When you're at 60 degrees, right? Your y value is your sine theta square root of three over two and your cos is one over two, right? It's the flip of 30 degrees, right? So this coordinate is one over, oops, one over two and your y is square root of three over two. So here we're at square root of three over two and here we're at, so now that we did the three trig ratios we found out what our coordinate system is in the first quadrant as we move along these angles. Let's take a look at what these angles are in radians and we've talked about this converting between degrees and radians and I showed you the long way to do it with the ratios especially you need to use that if you don't have special angles. The other way I showed you was multiples of pi, right? So that's what we're going to do to generate all the radians for all of these degrees. So what we're gonna do right now is generate the radian column since we're on it, okay? So zero degrees in radians is zero. You're at zero radians. We know that already, right? 360 degrees is two pi, we know that already. 180 degrees is pi radians, right? Half of 360 if you go full rotation. 360 degrees means two pi radians and we talked a lot about this already. That means 180 degrees is pi, right? So 180 degrees is pi. Now, in the first quadrants we can get to all of these angles 10 degrees from 180, right? So to find out what 30 degrees is in radians we look at 180 degrees, right? So we ask ourselves, what do we do to 180 degrees to get to 30 degrees? What do we do to 18 to get three? We divide 180 degrees by six, right? 180 degrees divided by six is 30 degrees. Well, the beauty of mathematics is very symmetrical, right? So you're gonna divide 180 degrees by six to get to where you wanna be, which is 30 degrees right now. Then we're gonna do the same thing to radians. So pi over six is 30 degrees. 45 degrees, what do we do to 180 degrees to get to 45 degrees? We divide it by four. There's four 45s and 180, right? You go one, two, three, four 45s, right? Well, if we're gonna do it to 180, we're gonna do it to pi. So 45 degrees is pi over four. What do we do to 60 degrees? What do we do to 180 degrees to get to 60 degrees? We divide 180 by three. We do the same thing to pi. So pi divided by three is 60, right? Pi over three. 90 degrees, we divide 180 by two, right? So we're gonna divide pi by two. So 90 degrees is pi over two. Now, all the other angles, 120 all the way to 330 are multiples of 30, 45, 60, right? Because that's what they are. We're moving around with the same displacement and the angles anyway, same angle changes, right? So that's exactly what we're going to do. So what we're going to do to find out what these angles are in radians, we're gonna look at the first quadrant and we're gonna multiply the angles in the first quadrant by something, right? So we ask ourselves 120 degrees. We can't get to 120 degrees directly from 180, right? Well, we can but it's a fraction. We wanna make the calculations easy, right? So we go to 120 degrees and we ask ourselves, what do we do? How do we get to 120 degrees from one of these angles, 30, 45 and 60? Well, we could multiply 30 by 40, 30 by four, that gives us 120, right? So that means in radians, we do the same thing. This would be four pi over six, but then we have to reduce the fraction. So what we do, we go to the biggest number that we have to get to 120 degrees. So 60 times two gives you 120. So we're gonna do the same thing to pi over three. So pi over three times two is the same thing as 120 degrees, right? So 120 degrees in radians is gonna be two pi over three. Now to get to 135 degrees, 135 is multiples of 45, right? And we multiply 45 by one, two, three, right? Just 345 is 135. So I'm gonna multiply pi over four by three. So 135 degrees is three pi over four, okay? 150 degrees, the only number I can use from the 30, 45 to 60 to get to 150 directly with one multiplication, one integer, right? It's 30 degrees, so 30 times five gives me 150. So I'm gonna multiply pi over six by five, which is five pi over six, right? Now 180, we already know what it is in radians, right? So that's just pi. So this column is just gonna be everything the same as this column. 210 degrees, I'm gonna multiply 30 by seven, right? Three times seven gives me 21. So 30 times seven gives me, so it's gonna be seven pi over six. 225 degrees is multiples of 45, and I always do this for 45. I never can do it by memorization. So 45 times one, two, three, four, five. 45 times five gives me 225. So five times five pi over four. So five pi over four is 125 degrees. 240 degrees, I get there from 60, okay? So six times four gives me 24, right? 60 times four gives me 240. So I'm gonna multiply pi over three by four. So 240 degrees is gonna be four pi over three. 270 degrees is 90 times three. So it's gonna be three pi over two. 300 degrees, I can get there with 30 and with 60. I'm gonna go there with 60 because that way I don't have to reduce my fraction, right? So 60 times five gives me 300 degrees. So 300 degrees is gonna be five pi over three. 315 degrees is multiples of 45, right? So I was at 545s when I was at 225. Five, six, seven, right? One, two, three, four, five, six, seven, 45s. So it's gonna be seven pi over four. And 330 degrees, I'm gonna get there using 30 degrees. 30 times 11 is 330. So 11 times pi over six, so 11 pi. And then we have 360 degrees, which is two pi. And one thing you should have noticed here as well is the symmetry within this, right? Because all we're doing is sort of doing a rotation like this. We're going to 90 and then we're coming back. So what's going on here is, so what's going on here is we're going to 90 degrees, right? And then we're going boom, boom, and then 180, okay? And then if we're here, if this was one column, we're 180 and then 270, sorry, 210 and 150 are multiples of pi over six. 135 and 225 are multiples of pi over four. 240 and 120 are multiples of pi over three. And then we're at 270, right? 270 on either side, we're multiples of pi over three, multiples of pi over four, multiples of pi over six, okay? There's a lot of symmetry within this table. That's why it's really easy to generate. And it's not really about memorization, it's about appreciation, right? It's about generating it. And the reason this symmetry exists is because let's do 90 degrees here. If we're at 90 degrees, 60 degrees and 120 degrees are going to have the same magnitude up on the x-axis and the same magnitude along, sorry, same magnitude up on the y-axis and the same magnitude along the x-axis, right? So if we're sitting here, if we're sitting here and this is 120 degrees, right? Our x-value, our y-value is again, three over two, right? That's our horizontal, we're right there. Our x-value, if you come down, that's going to be the same distance along as it was in the positive direction, right? So our x-value here becomes one, two, three, four and a half. Our x-value here becomes one over two, it just happens to be negative, right? Because we're in the negative x quadrant, right? So our coordinate system here is, our x is negative one over two and our y-value is root three over two, okay? My red pen is dying. Let's see if we've got a better red pen. When we're at 45 degrees, we're here, our y-value is going to be the same, right? If we're standing here and our x-value is going to be the same distance along on the positive x-axis as it was in the negative x-axis, right? So if our x-value here is going to be one over root two, but negative, so our x-value is negative one over root two and our y-value is one over root two, right? That's the symmetry that we see. When we're standing here, when we're going across, we just went 30 degrees on either side of the 90-degree angle. So our y-value better be the same and our x-value is the same, right? The next one is 45. We've gone 45 on either side and then we're here. Our y-value is the same and our x-value is the same. Next one is 60 degrees on either side of the 90-degree angle, right? Well, 60 degrees is going to bring us back to 30 and it's going to bring us to 150 degrees, right? If you see it here, you're at 150 degrees, okay? Oh, sorry, right here. We're at 150 degrees, right? Right here. And that's the beauty of the symmetry, right? And our x-value is going to be negative square root of three over two. I'm just reading it off here. I'm just making it negative, right? And our y-value is going to be one over two. So this is negative square root of three over two and that's one over two. And that's what we're going to do right now is generate these guys, right? So the symmetry that exists here, as soon as you come here, you hit 90 degrees, four days guys, they repeat, you go back. So sine of 120 degrees is root three over two. Sine of 135 is the same as sine of 45, right? Sine of 135 degrees, you're here. Well, that's going to be the same value as sine of 45 degrees, right? Same y-value. One over root two, she's lose. One over square root of two, right? Sine of 150 degrees is going to be the same as sine of 30 degrees, so one over two. And then when you're at 180 degrees, that's zero. So when you're 180 degrees, your y-coordinate is zero and your x-coordinate is going to be negative one, which we're going to take a look at, right? So right here, we're at negative one and zero, okay? Let's do cos, well, cos does the same thing. The symmetry exists from each quadrant to the next, both, well, for sine, cos and tan, right? So as soon as we generate the sine, we can generate the cos for the first quadrant and we can generate the tan. And then once we got the first quadrant here for the three trig ratios, we have the rest because it's just mirror of each other. So as soon as you hit zero, so you're coming along, you go, right? 30, 45, 60, 90, and then you go back again. So the mirror point on this is going to be a half, one over two, and three over two, right? And one, the only catch is that you have to remember is when you're in the second quadrant, your x-values are negative. That's why these guys are negative, right? So your x-values are negative, their magnitude is the same, their absolute value is the same. So all you do now is make this negative, negative, negative, negative, okay? The tan, same thing, same symmetry. You're at 90 degrees, on either side is gonna be square root of three, one, one over square root of three, and zero, right? And since tan is sine divided by cos, right? Then positive over negative is negative. So the tan values here are also negative. Negative, negative, negative, okay? As for the remaining stuff here, well, let's draw our circles, put our little dots. One, two, three, four. I'm just gonna line it up with this. When we're here, we're gonna be here, and we're gonna be here, right? I'm just lining them up, right? Our x-values are gonna be the same, and our y-values are gonna be the same as these guys, but they're gonna be negative. So if I come over, one, two, three, four, a bit, oops, let's get you around here. Negative, a half, negative, one over square root of two. Negative, negative, root three, over two. Two. So it repeats again. It does a rotation, right? So all I do with this is, when I'm doing the signs, when I come to one, I go boink, boink, boink, boink. So I'm at zero. I'm just gonna copy this column down here. Zero, negative one, and zero, okay? So when I'm here, again, this guy repeats. So it's just gonna be boink, boink, boink, right? A half, right? It's just gonna be these guys, so one over two, one over square root of two, root three over two, and one, right? But this is in the third quadrant, and we're talking about signs. Sign happens to be y, so we're in the third quadrant. We moved along here, now we're down here. Sign is negative here, so all of these numbers are mirrored, but they're negatives, right? Because sign is negative. And then when I get here, I do the same thing, I go back again, right? Because what I'm doing, I'm standing here, and then I'm coming back up again, right? So I'm the same distance away as this guy, right? There's the same distance here, okay? So this guy is gonna mirror again. So 300 degrees is gonna be same as 240 degrees for the y value, right? 240 degrees, I'm here, my y value is this, 300 degrees, I'm here, my y value is this, right? So again, it mirrors. So negative square root of three over two, negative one over root two, negative one over two, and zero, right? And again, it's negative because sign is negative in this quadrant, right? As for cos, well, that was a negative one, and I go back again, right? Which is basically the same as this. So it's gonna be root three over two, one over root two, one over two, zero, right? When I get to zero, I'm just gonna mirror it again. One over two, one over square root of two, root three over two, and oops, one, and one. Now, cos, this is in the second quadrant, right? Oh, sorry, third quadrant, right? Cos is still negative, so my x is negative, so my cos is gonna be negative. This is in the fourth quadrant, I'm over here now, and my x-axis is positive, so my cos is positive. 10, same deal, I'm just gonna write these down. I hit zero, and I'm gonna go back again in this way, or I hit zero, so I'm gonna go this way, right? One over root three, one root three, undefined, and then mirror again, root three, one, one over root three, and zero, right? Now, all I have to do is make sure it's positive, you know, figure out which one's positive, which one's negative. 10 is sine divided by cos. If sine is negative, cos is negative, then they're negative. Over here, sine is negative, cos is positive, so 10 is negative, okay? And the values here would have been, if we're drawing this, might as well put our little guy here, right? So we're here, we're here, I guess, yeah? And we're here, right? And these guys line up with these, right? And that's us moving around the unit circle, right? Generating table, and figuring out what our coordinates are, our x and y's are, where sine and cosine are as we move around the circle, right? Now, I guess I should put those things there, but my red pen is dying off, so I'll leave those guys empty, okay? And this is something that we have to learn, we have to know how to generate. And if you're writing a test for me, if I was giving you a test, the first question on my trigonometry test would be generate this table. And what you would have to do is you would draw this, the unit circle, you would draw that, and you would draw that. This is the only thing you need to be able to generate this. Because from this, you get your degrees, you get your sine, and once you have your sine, you can get your cos, because that's the mirror along 45, or you can just read it off, the special right triangles and your unit circle, right? You can get your tan, and we get our radians from the pi, from the ratios, right? And just a little note, if you wanna remember this, or understand it, anything that you see that has pi over six is a multiple of 30 degrees, right? Anything you see that has pi over four is a multiple of 45 degrees. Anything you see that has pi over three is a multiple of 60 degrees, and anything you see that has pi over two is a multiple of 90 degrees, right? So if we come over here, and we see pi over four, then pi over four is a multiple of 45 degrees. So seven pi over four means seven times 45. And what we're going to do in the next video is do one generate this table with commentary, but go through it much faster, right? I'm not gonna explain all this stuff. I'm gonna assume we know what these numbers mean now, right? So you basically what your coordinate system as you move around the circle, right? And then the second part of it, or in the next video, is going to be us doing it, generating a table without any commentary, as if you would on a test, right? Because as soon as you get your test, you're gonna write down your name, put that test aside, grab a scrap piece of paper, and generate this table, hopefully it'll take you less than two minutes to generate this whole thing, and you'll have that table beside you to answer a lot of questions during the test, right? It's gonna provide you answers for a lot of it, right? Because for example, let's assume a question would ask you, find theta in radians for when sine theta is equal to negative one over two. So all you have to do now is go to your sine theta, go across, find negative one over two. Oh, we got one negative one over two and another negative one over two. So the angle and degrees of sine theta for negative one over two is either seven pi over six or 11 pi over six, right? Easy peasy. It's one of the first types of questions you get on a test, right? Or they could ask you, give the three trig ratios for, I don't know, five pi over four. So you would come down here and go, oh, sine of five pi over four is negative one over two, negative one over two for cos and one for tan, right? They can ask you to convert 150 degrees to radians. And all you would do is come up here and go 150 degrees is five pi over six, right? We end up using this a lot, okay? In the next video, we're gonna generate this and then after that, there's two different pathways we're gonna go down. One of them is graphing trig functions, using radians mainly because we've graphed them in one of the first videos we did just as a teaser, just to see what these graphs looks like. The other path we're gonna go is taking a look at identities and seeing how identities come into play and how we can manipulate the relationship between these ratios and what we know of right angle triangles and circles and come up with exact values for other things other than just these angles, right? Other angles, right? And the way you should think about this is this basically is us creating a whole universe, right? Creating a system that explains many other systems within our reality, within our everyday lives, right? As for why this is important because cyclic functions are dominant within our experience of life, right? Within a lot of systems that we function under as well as how we perceive life, right? How we perceive reality, right? And what we're going to do is manipulate the basic cyclic function that we have which is the unit circle, the simplest wave we can think of and we're gonna see what other types of waves, what other types of systems we can take a look at. I'll see you guys in the next video. Bye for now.