 What we're going to look at today is, in fact, for about four days, and it's the final lesson, is graphing trig functions. Mostly we're going to be graphing sine and cosine. We'll do tangent a little bit, but as you'll find out tangent is yuck. Sine and cos, tangent, and then we'll look at the reciprocals. But sine and cosine and tangent all belong to a certain type of function called a periodic function. A periodic function is a function whose graph repeats regularly over some interval of the domain. The length of this interval is called the period of the function. And those of you that are in Physics 12 wrote, oh yeah, once around the circle was the period. And then it's going to give you another term. The term is amplitude and it defines amplitude as half the distance between the maximum and minimum values of the function. Say, oh, it's much easier to look at graphs and figure it out. So we have three graphs here and we are going to write the amplitude and the period, the amplitude and the period, the amplitude and the period. The amplitude of this first graph here, Emily, is 10. How high does this graph go? 10 up, 10 down, 10 up, 10 down, 10 up, 10 down. That's what's meant by the amplitude. In fact, Spencer, this graph is centered right around a height of zero, which makes it easy to spot the amplitude. Trevor, the next graph is not centered around a height of zero. And this is where we're going to use what they've said here, but here's how I find the amplitude. Trevor, how high does this graph go, the first graph? Look up, how high does the first graph go? How low does the first graph go? What's the total distance between the top and bottom? Divide that by two, that's your amplitude. You ready? How high does this graph go? How low does this graph go? What's the total distance between the top and the bottom? Eighth, divide that by two, that's your amplitude. This graph goes four up, four down, four up, four down, four up, four down. As a matter of fact, Maria, that means if I want to find the middle of this graph, it's four down from the top or four up from the bottom. This graph is centered right around a height of positive two. Eric, how high does graph C go? Sorry? Six, how low does it go? What's the total distance between them? The total distance between them? Half of that. That's your amplitude. Period. Period is how long one wave is, how long the graph goes before it repeats itself. When I find the period, Brett, I almost always try and do it with the easiest points possible. I almost always go from peak to peak or from trough to trough. In other words, if you say to me find the period, from here to here, how far along the x-axis is that? Period's eight. You may notice, Ellen, I'd get the same answer from here to here, six to fourteen, still eight. In fact, I get the same answer, Mitsu, from right here to right here, but that would be stupid points to pick because I'd be making my life way tougher and I'm increasing the likelihood of making a mistake. If I have a choice, I pick from one peak to the next peak or from one trough to the next trough. So Maria, if I want to find the period for B, how far is that distance? Six. If the graph has a period of six, it repeats every six units. Asar, C is much more difficult, but I would go from the left-hand side of one peak to the left-hand side of one peak. I think that's where it starts to repeat. How far is that? Ten. It's got a period of ten. I could have also gone from the right-hand side of one peak or from the bottom of one. Anyways, be consistent with that. We're not going to be looking at A and C very much. Graph C is called the heavy side function. It's actually the physics of what happens when you hit some of the things. Sledge hammer. No force. Sledge hammer, all the force maxes out. There's your follow-through. No force once the sledge hammer stops moving. But B, this, this, my children, is a trig function. Turn the page. We're going to graph Y equals the sine of X. Except it says where X is in degrees. No, no, I say no, no. We have left degrees behind. It's not like you all cross out the degrees symbol and cross out the 360. Zero radians is zero degrees. I don't need to change that number, but 360 degrees, how many radians? Let's put a 2 pi here. You have to memorize the sine graph. Except what I'm going to teach you today is how I get away with memorizing only bare-bones basics but can get all of the information in one second. I'll start up by saying, you know what, I'm not going to label all of these points. Too much work. I'm always interested in here, here, here, and here. In particular, I'm interested in zero radians, 90 degrees, which is how many radians? Pi by 2. 180 degrees, which is pi radians. And 270 degrees, which is 3 pi by 2. 360 degrees, which is 2 pi radians. I'm going to fill those in, and the way I'm going to fill them in is to use the unit circle. Jimmy's sine is whatever what in terms of x and y are. Louder with authority, y over r. Now on the unit circle, we specifically said let the radius be 1, which means if y over r is sine but r is 1, then sine just becomes plain old, your y coordinate. How high you are. So here's my unit circle. There's zero radians. How high am I right there? Zero turns out the sine of zero is zero. The first point on my graph goes through zero. The next one I said I was interested in was pi by 2 right there. Isabel, how high am I right here? 1. You know what the sine of pi by 2 is? 1. You know what the sine of pi by 2, now pi by 2 on here is 90 degrees? 1. The next point that I would do would be this bad boy here. How high am I at pi? Zero. 180 degrees, I'm zero high. The next one that I would do is this one. How high am I right there? Careful, not one. Negative one. Oh, I forgot to fill in the zero for pi. Negative one. And then we're back to where we started from with zero. And the nice thing is the sine function is periodic. In fact, it goes zero one zero negative one zero one zero negative one zero one zero. It goes top middle bottom middle top middle bottom middle top middle bottom. It's a nice periodic function, which means I can actually ditch the math. Oh, how far apart is each dot? I've watched that 360, didn't I? Sorry, my 360 is in the wrong place. Let's try that again, Mr. Deep. Right there. How many squares apart on this graph is each dot? How many squares till the next dot? Three? One, two, three squares. How high will I be? Continue the pattern at 450 degrees. How high will I be? Yeah. What about at 540? Zero. Hey, let's go backwards. What about at negative 90? How high will I be? Negative one. Negative 180 is zero. Negative 270 is one. Negative 360 is that. So plot the points on the grid below. Do not join the points. We have plotted the points. Now we're going to go to our graphing calculators. Get your graphing calculators out. If you don't have your graphing calculator here, you're going to be officially fessed up because you're going to be using it during this whole class. So get your graphing calculator out and go y equals. Clear whatever graph you have there. Also, make sure that your plot one, plot two, plot three are not accidentally highlighted. If they are, go up arrow until you're on one of them and press enter to de-highlight it. And we're going to go sine x equals bracket. Oh, wait a minute. They did degrees down here. So let's go mode and let's go into degrees one last time temporarily. But I want my graph to look exactly like theirs. So before I hit graph, if you did hit graph, that's fine. But I'm going to hit the window button and I'm going to match this to their grid. Caitlyn, how far left does their graph paper go? There's my y-min, negative 360. How far right does their graph paper go, Caitlyn? Let's put that in there. What's each square worth on the x-axis? I'll have to do a bit of math, but I notice three squares make up 90 degrees. So what's one square worth? 30 degrees. I think their scale is 30. Yes? I also noticed vertically that five squares make up one. So what scale are they using vertically? What's each square worth vertically if one makes up five squares? Point two. And I think that means that the lowest it gets is negative 1.2. The highest it gets is positive 1.2, scale 0.2. And now hit graph and you will see the sine curve. The sine curve looks like this. If you didn't get that, now is your chance to ask. It's periodic. It repeats itself. It has an amplitude. Let's first of all copy it out. It's a smooth curve, so it looks like this. And I'll put arrows on the end to simplify. Hey, it keeps going. Pretty curve. You have to memorize it. I'm going to show you a fairly easy way to memorize it without truthfully memorizing every little detail. I'm big on it. If I can derive it in one second, I'll take the one second of work and save my brain. Let's look at this. Specifically, let's fill in the bottom here. What's the domain of sine? All reals? Oh, that's kind of nice. By the way, this is what some of the information you'll need to memorize. You can write down next to your memorize. Except, again, I'll show you a way you can derive some of it too. But still, it's worth having here. What's the range? How low does this sine graph go? Negative 1? Oh, that's not bad. How high does this graph go? The range is between negative 1 and positive 1? I mean, oh, as ugly as it could have been. That's a nice range. Negative 1 less than or equal to y, less than or equal to 1? What's the amplitude? How high does this graph go? 1? How low does this graph go? Negative 1? Total distance? 2? Half of that? What's the amplitude? 1? Oh, I mean, the only better than that would have been 0, but 0 would be a straight line flat line. That's a great amplitude. This is a nice graph. What's the period from peak to peak? Do the math. How many degrees from here to here? Sabina. Period. Oh, you say 306, once around the circle. Or in radians, 2 pi. That makes sense. We'll come back to the x-intercepts in a second. Let's do the y-intercept. What's the y-intercept? What, comma, what? Ellen. The y-intercept is 0, 0. That's, I mean, as ugly as it could have been, that's what Mr. Dewick's asking me to remember. Hey, I like this. Why do they have x-intercepts in brackets? Because there's more than one. In fact, there's an infinite number of x-intercepts. I can't tell you all of them, Vlad, because they keep going and going. What I'm going to do is I'm going to describe the pattern. Don't write this first bit down. Don't write this first bit down. I'm going to do it the long way, and then I'll show you the shorter way that we write it. Where is my first x-intercept, Rhett? 0. How far till the next one? How far till the one after that? How far till the one after that? Okay, here's how I'm going to write this. But I'm going to shorten it down, so I don't write this down yet. Starting with 0 degrees, add or subtract multiples of 180 where n is an integer. In other words, 0, 180, 2 times 180, 3 times 180, 4 times 180, 5 times 180, 6 times 180. Except this isn't quite right because, Isabel, what's 0 plus anything? Does the 0 actually make a difference? What's 0 plus 5, Caitlin? What's 0 plus 6? What's 0 plus 180 n? They generally don't put, because you're starting with a 0, they don't put that in front. If we were starting somewhere else, we would. Write that down. Oh, and let's do the radian version. Pi n, where n is an integer. This is the sine graph. It follows a pattern. Middle up, middle bottom, middle up, middle bottom, middle up, middle bottom, middle bottom, middle bottom, middle bottom. It actually corresponds to here, here, here and here. Has an amplitude of 1. Domain all the yields. Turn the page and look at assignment homework question 1. Now it wants us to look at cosine. Once again, we're only going to look here, here, here and here. Once again, we're going to use our handy-dandy little unit circle. Cosine, mitzuh is what over what? And if r is 1, cosine ends up just being how far left right you are, your x-coordinate. So, what's the cosine if we start at 0 radians? What's the cosine of 0? What's your x-coordinate right there? You're telling me my x-coordinate is 0? No, that would mean I was on the y-axis. Sorry. Cosine of 0 is 1. This graph starts out going through 0, 1. The next one I'm going to do is right here. It's pi by 2. Spencer, what's my x-coordinate right there? So, the cosine of pi by 2 is 0. Now, I've got to improvise a bit on my graph. Oh, wait a minute. Six squares is pi. Where would pi by 2 be? How many squares? Three. I'm 0 high. Three squares in. Next, what I'm going to do is right here. Pi. What's the cosine of pi? What's my x-coordinate right there? Do you say negative 1? Thank you for getting the negative. The next one is 3 pi by 2. Right there. What's your x-coordinate right there? 0. And that would be three squares over again. Oh, I forgot to put the 0 in here. And the last one, back to where we started from 2 pi, what was the cosine of 0 or 2 pi? And I've just begun to repeat. Three squares over, 0. Three squares over, bottom. Oh, I can back count as well. Three squares left, 0. Three squares left, bottom. Three squares left, 0. Three squares left. The cosine graph looks like that. Looks like that. In fact, Emily, I'm going to argue for what it's worth. If you took this sine graph and you just slid it 3 left, it's exactly the cosine graph. In fact, that's how sine and cosine are related. Or, for that matter, if you moved it 3 right, I think. Nope, never mind. 3 left. So, turn the page if you haven't. 278, number 2. Let's see if we can fill this in. Because you need to memorize this too. What's the domain of cosine? All reals just like sine? Really? Oh, that's too good to hope for. Next thing you know, you're going to tell me that the range is between negative 1 and positive 1 just like sine. You're not going to tell me that, are you? You're kidding me, really? Oh, great. I would almost pay money for math like this. Don't tease me now. Don't tease me and tell me the amplitude is 1. That would be just too good to hope for. Really? Oh, man. Well, I will lose it if the period ends up being 2 pi or 360 degrees. Is the length of one wave 2 pi or 360 degrees? Oh, you're kidding me. Man, I need to update my Facebook stat. This is great. In love with cosine. You know where this breaks down a little bit, though? The x-intercepts and the y-intercepts are different. But once again, let's do the y-intercept first. What's the y-intercept of cos? What, comma what? Zero? Like, it could have been anything ugly. And it's, I mean, 00 is the best we could ask for. But I think the next best we could hope for would be 0,1. That's not bad. What about the x-intercepts? Once again, Carly, I'm going to have to do a list. I'll mention the first one, and then I'll say, how far apart are they? The first one is right there. How many radians is that, Carly? How many degrees? 90. And the next one is six squares, which is pi or 180. Oh, you know what? They're still 180 apart, just like sine and cosine. They just start in a different place. Oh, so x-intercepts. I can either say pi over 2 plus multiples of pi or 90 degrees plus multiples of 180. Which one is correct? If you want me to do it in radians, I'll do it in radians one. If you want me to do it in degrees, I'll give them the degrees one. Which one would you want? The radians one. Cosine and sine. You know what they're like? They're like, if you ever see two siblings who are a couple of years apart that aren't twins, but boy, they could be twins. There's only a few things that make them different. Andrew and Brody. So it's who I used last class. How many of you come from a big family of five or more kids? Not very many. Both my folks do. Which means I got uncles and cousins galore. I got cousins I've never met. In fact, I played basketball and was friends with a kid who told me I was about 21 or 22 before I figured out he was actually my cousin. I assumed he was going out with one of my cousins and that's why I kept showing up at my family gatherings. I didn't know. Different last name and... Big families. And my grandpa remarried after his first wife died into another big family and was like, holy smokes. Related to half of Abbot's room. As soon as you get a big family, maybe you guys can relate to this. In my family, we also have the ugly cousin, the one that doesn't quite fit in. For us, it's cousin Marvin. Marvin was very cool when I was eight years old, but now he's in and out of jail all the time. He's a biker. Where is he? He's somewhere in the Lower Mainland. How many kids? We don't know. What job? We don't know. My family gatherings late on his Harley and seemed very, very friendly. Now I know he was back then. He just seemed very, very funny to a little kid. Tangents, the ugly cousin of the Triggs family. Tangent, unfortunately, is not going to fit in at all. At all. Turn to page 279. Sign and coasts really are as clean Tyler as math can get. Same domain, same range and a nice domain on range. All reels and plus or minus one. Same amplitude and a nice amplitude one. Nice X and Y intercepts. Nice. Tangent. In fact, tangent, I'm not even going to go to the unit circle because even the unit circle kind of breaks down for tangent. Tangent, I'm going to fall back on my most fundamental definition, X and Y and R. Tyler, tangent is what over what in terms of X and Y and R? I'm going to write that over here. And as soon as I write that, there's a problem with this fraction because this fraction can sometimes be undefined, which is why I'm so glad someone asked that earlier. What would make this fraction undefined? When what's zero? When X is zero. Here is my circle just so I can kind of get my bearings. Where is X zero? Right there and right there. Where is tangent undefined? Right there and right there. How many degrees is that? 90, how many radians? Pi over 2. The graph gave me as in degrees at 90 degrees. There's going to be a great big undefined vertical asymptote again. This is how I draw tangent, by the way. Now I also know going around to 270, I don't want to go that way. Instead, I'm going to go this way. It's also undefined going this way. If I go in this direction, Sandaly, how many degrees is that? There's going to be an asymptote at negative 90. This is what's going to give me centered at zero. So negative 90. Always when I'm doing tangent, I almost always will draw the asymptotes first because they're going to be my guide railings. Then, in all honesty, I try and get one more point if I can. Three more is nice and because this is the basic tan, we'll actually get three. Otherwise, I just need one. What would make this fraction work out to zero? What makes a fraction equal zero? Not when the bottom is zero, that's undefined, but when? When the top is zero, when Y is zero, where is Y zero? Right there. What angle is that? This graph goes through zero, zero. Actually, you know what? I don't derive that part. That part I've just memorized. Tangent and sine both go through zero, zero. Then I fall back on one more nice angle, but it's actually not from the unit circle. It's from this triangle here, the one, one root two. What angle is that right there in the one, one root two and since we're in degrees, I guess we'll go with degrees. 45, what is the tangent, the opposite over adjacent to 45? I know that tan of 45 is one and tan of negative 45 is negative one. Those are the two extra points I try and get if I can for my total of three. Now 45 degrees, I'm not really fond of the scale here, but okay, if that's 90, 45 is going to be one and a half squares over where one high and negative one and a half squares over where negative one high. And tangent looks like this. Nothing at all like sine or cosine. In fact, it looks a little bit like this. By the way, that's one complete tan graph. It's not two pi or 360 degrees period. How wide is it? How many degrees is the period of tangent? 180 pi, which means it goes twice as that. It means if I want to draw the next one, I would say, well, the next asymptote is going to be 180 degrees over, six squares right there. Oh, I went too far. Don't try that again, Mr. Duk. One, two, three, four, five, six squares right there. It's going to go through zero, zero in between the two asymptotes. Halfway along, it's going to be one high. Halfway back, it's going to be negative one high. It's going to look like this. Going in the other direction, six squares back, 180 degrees or pi radians. One, two, three, four, five, six, three. Going to be another asymptote right there. Once again, exactly halfway between the asymptotes. That's where it's zero, zero. Halfway to halfway, it's going to be one high. Halfway to halfway, it's going to be negative one high. I guess I'd also have that and that if I wanted to really fill in another half of a wave. Tyler, these points are, these two are really optional because often you won't be able to get that one high and negative one very easy at all. The center point is the key one and the asymptotes are the key one and the shape is the key. Even your graphing calculators have a really, really tough time with this. If I go tan x, you may notice that what I get on my graphing calculator doesn't exactly look like what we drew. It really has a tough time with those vertical asymptotes. It wants to put them in as lines. Sometimes, although you'll notice in the middle there, it figured out it was a vertical asymptote because it didn't put a line right there even though it put one everywhere. The ugly cousin. Now, on all of your graphing calculators, when you hit the zoom, one of the options is zoom trig. If you pick zoom trig, it will pick a scale usually from negative 360 to positive 360, negative 2 pi to positive 2 pi. I think there the software knows not to put asymptotes in and mix it. That's what the tandem graph is supposed to look like. You can't trust your calculators here. What was it? Negative 360 and 540 and negative 1.2. 1.2 I think is what we had. The other really big issue is if you're trying to solve an equation. Suppose I said solve tan x equals 3. I better make sure my window is 3 high. Let's make it 4 high. I would solve that by graphing that, if I wanted to check out my graphing calculator, graphing that and then finding where these two lines cross. The problem is my graphing calculator might also think that they cross right there even though there's nothing there. So you have to look at your solutions where you're hovering in the air very carefully. Did I mention tangent was the ugly cousin? Yeah, the ugly cousin. What's your homework? Number 4 and then oh, wait a minute. We haven't finished. We're going to do number 7 together. What's the domain, Kara, when you're done yawning my child? What's the domain of tangent? Tangent does go forever. It is sort of all reals except there's gaps. So rather than say everything that it is, it's easier for me to say what it isn't. For the domain, I'm actually going to say x can't be. What can't x be? Where was the first asymptote that appeared? 90 degrees. How far till the next one? 180. x can't be 90 degrees and then multiples of 180 degrees. That's if we want the domain in degrees. If we want the domain in radians, x can't be, all right, 90 degrees radians. Pi by 2 plus 180 degrees multiples of pi where n is an integer. This may be your first experience where instead of saying what the domain is, we say what the domain ain't. Isn't it easier? What's the range? All reals. That's about the only easy thing of tangent. And because the range is all reals, there is no amplitude. How high does it go, infinity? How low does it go? Negative infinity. What's infinity plus negative infinity? How far apart are they? Two infinity. What's half of two infinity? Infinite. Oh, okay. Yeah, no amplitude. Ah, there is a period though. How long was one wave? Not 360. 180 or pi? We'll do the x-intercept second. It does have a nice y-intercept. Built for 0, 0. Okay. What about the x-intercepts? There are many. Where is the first one at 0? How far till the next one? 180 degrees? Oh, let's do it in radians. Well, let's do it in both. So, x-intercepts, it's going to be 0 degrees plus 180 degrees times n where n is an integer. Although, you know what? They wouldn't write the 0 degrees because 0 plus 180 is just 180. Pi n where n is an integer for radians. Tangent is equal to 0 at 0, 180, 360. Oh, heck, it's easier to do it in pi. 0 pi, 2 pi, 3 pi, 4 pi, 5 pi, 6 pi, 7 pi, 8 pi, 9 pi, 10 pi, 11 pi. Tangent is 0 there. Negative pi, negative 2 pi, negative 3 pi, negative 4 pi, negative 5 pi, negative 6 pi. The last thing I'm going to ask you on your test is the equations of the vertical asymptotes. These red vertical lines, I might want to know their equations. Now, there are many of them. It's going to be easier for me to describe one and then say when the next one appears. Quick math 10 review. What's the equation of that vertical line that goes through x equals 4? And I just told you, x equals 4. That's the equation. What's the equation of the vertical line that goes through x equals 4? x equals 4. What's the equation of a vertical line that goes through x equals 90? When's the next one? How far till the next one? Equations of the vertical asymptotes, 90 degrees plus 180 N or pi by 2 radians plus pi N where N is an integer. In fact, you may notice that the domain what x can't be is the same as the equation of the asymptotes what they are equal to. You can also add 8, 9, 10 and 11 to your homework. And then I promised you guys that I would show you an easier way to remember sine and cosine. I got not really good for tan. How do I do tan? I draw the asymptotes. I know it goes through zero dead center between the two asymptotes. I can usually figure out the asymptotes by saying when am I dividing by zero? That's where the problem is. And then I know it looks like C. But for sine and for cos, can you scroll down to where the answers are? And I think at the bottom of the second page of answers, we've got about a third of a page. Here's how I do y equals sine x. I'd like you to write down y equals sine x. But then I want you to put your pencil down, watch me do the whole thing. And then what I'm going to do is say now try it yourself from memory to help. Here's what I do. Someone says to me, hey, let's talk about the sine graph. I go, well, once around the circle, I do that right away. Don't write this down yet. Justin, what does that have to be then? See how you can get that without memorizing it? What does that have to be then? See how you can get that without memorizing it? What does that have to be then? Come on, do the arithmetic. Thank you for not letting me down. I do remember that sine started here. And then I know it goes middle up, middle bottom, middle up, middle. Oh, I add one high and negative one high. I do have to remember that. But is one and negative one, Trevor, fairly reasonable to remember? I think so. And I know it goes like this. And when I connect them, if you all turn your head sideways, what letter does sine look an awful lot like? S. That's when someone says to me anything about the sine graph, this is what either pops into my head or Alex on a piece of paper. I derive it in. It takes me one second far longer than it took me to describe to you. Try recreating that yourself. Hey, what's the sign of three pi by two, negative one? See it? This also, if you're a little confused on the unit circle thing that I've been doing for the alarm bell questions, I also suggest if you don't like the unit circle, you can sketch that in all of one second. And that'll take care of where sine is negative one or zero or positive one and ditch the whole wrong unit circle. I do the same thing. And again, put your pencil down and watch. And then try recreating it afterwards. Do the same thing for cosex. I sketch one period, which is once around the circle, which is two pi. Then it's pretty easy for me. Assar, what's that have to be? Gotta be. What's that have to be? And I know it goes one high and negative one high. And I've memorized, cosine goes through, starts at one, starts up high. And then it's going to be zero, bottom, zero, top. The cosine graph, turn your head sideways. It sort of looks like a C. Not quite as nice as the S, but it sort of looks like a C for cose. I remember. What do you got for tangent, Mr. Dewick? Nothing. Well, what letter does, well, write this down first. This is the best I got, and it's really not much. What letter does tangent start with? Now I'm a T, now I'm a tan. It's sort of a T, if you imagine my body now gone, but the x-axis or the y-axis are sort of that there, which is the tangent graph done with the computer, sort of looks like if the y-axis was the main part of the letter T, kind of ugly cousin. Got nothing, really. So what's your homework? You have a take home quiz and you got a few questions.