PC 10.1 Graphing Sine and Cosine

Hey guys, welcome back to another lesson in precal. This is Mister Bean, and today we're gonna be talking about graphing sine and cosine. I know graphing is your favorite part of all of math, so I know you're excited for this one. The graphs of sine and cosine, which mister breast introduced to you, are going to look like waves. And we call them sinusoids or the way I remember them, they are like sick dinosaurs, get it sinusoids . There's a little bit of stretch there, but I like that. Sinusoids are the graphs of sine and cosine. Lots and lots of applications are involved with sine and cosine. You've got soundwaves. You've got light waves, some different cool things or how the light reflects and separates. You've got anything dealing with springs that has sine and cosine graphs, pendulums, microwaves. That's a clean microwave. Here's a picture of brust microwave. See all the students who use it in his room. It's really disgusting. And then we've got ocean waves, and earthquakes. Basically, anything that would have a repetitive cycle to it is going to have some involvement with sine or cosine if it repeats over and over again. So lots of applications. So how do we figure out the graphs of cosine and sine if we have no idea what they are doing? Well, let's think about using the unit circle where cosine X will be the X value of whatever angle we're plugging in. So for example, if I say 60° or pi over three, same thing, then cosine of pi over three is one half. Sine of pi over three would be square root of three over two. So the sine is the Y value of the cosine is the X value. So if we're looking for the cosine value, that's all of the X's. So we're going to look for all of the X coordinates to get cosine and fill out this chart. So when the angle is zero right here zero radians, then the cosine right there, the one, that's what we put one. And now let's go to pi over two. The cosine value is right there. It is a zero. Over here at pi, so we go our angle of pi, the cosine value is negative one. And then three pi over two down here, cosine value is a zero. And then the last one, two pi, so we come out here all the way back full circle, and it is one again. So what are the, what's the cosine graph doing? Values of cosine. We started at one, it got a little bit smaller, a little bit smaller down to zero, and then what are they doing? It's going negative negative negative to negative one, then a little bit less negative, less negative back to zero, and then a little bit larger back to one. So what's just going from one, zero, negative one, zero, and one around in a circle over and over again. That's what it's doing. So let's jump ahead to that part of your notes and graph cosine X the first coordinate point is zero comma one. So if you look at that T chart, here's our first point. Now, this is very important to understand. If you can't remember where cosine graph starts, it's going to start right here at zero one. I'll get back to this in just a second. Now what's the next coordinate point? We have pi over two and zero. So I'll put that there. And then we had pi negative one. So I'll put the dot right there on top of the pie on your graph because that's your negative one. Three pi over two and zero. So you can see here, all I'm doing is I'm just plotting the points from our little T chart, right? Up above. And then if you can see the pattern here, I could work backwards and you'd have another X intercept. And you can see a little bit of what our graph is doing. Now be careful, don't just go sharp these like this. That's not what our graph is doing. What you want is to create some nice curves here for some oh, that was not very good. Let's try that again. So we're going to create some nice curved maximums. And some nice curved minimums. Nice curve maximum. And then it's going to cross through the X intercepts. There, whoops, there we have our cosine curve. And it just keeps going in the same pattern, making some waves like this up and down, up and down. So let me say that one more time again. When you start with cosine graph, you are starting right here with a maximum. Let's write that down in fact. Just put that in your notes. Starts with a maximum, okay? It might also start with a minimum, but I'll show you how later in the notes where that might happen. But that's the big thing. Cosine starts with the max. And let's go back and fill in the sign graph. So let's figure out what's the Y value. So remember sine is a Y value of all of these points. So when the angle is a zero, 0° or zero radians change that, make it so we're focusing on the Y value. So now we're focused there on the Y value, and we have a zero. So sine of zero is zero. Sine of pi over two is one. Sine of pi is zero. Sine of three pi over two, remember we're looking at the Y value. That is a negative one. And then we come back in a full circle of the Y value right there on the coordinate circle is a zero. So now let's plot these points. The starting point for sine is at the origin because we go through the coordinate .0 comma zero. Now let's keep using that T chart and fill in the rest. Pi over two, and one. Pi we're back to zero. Remember we're doing the sine values. Three pi over two will be at negative one. And then the Y value on the unit circle at two pi is back to zero. So just like cosine, it will continue to follow this pattern. There's our negative one. And then we'll fill in some nice curved maximum, a nice curved minimum, and then here as well, nice curved minimum. And kind of connect those through the X intercepts. Okay. And then the sine wave continues and it continues on forever and ever and ever. So what we're going to say about the sine graph, the sine graph, starts at the origin very important, sign graph starts right there at the origin. And then how long does it take until it repeats? It's going to repeat itself way over here when we get out to two pi. Just like cosine, cosine graph starts here, and it repeats itself when we get out here to two pi, the cycle starts over with a maximum. For sine, it starts over going, starting to go up. And then it goes down, and then on its way back up, it is now completed one cycle or one circle. On this part of your notes, we've got a few definitions that have already written out there, but make sure you get that the amplitude is the absolute value of a, don't just put a, it is the absolute value. So a mistake that kids will make would be that if it says Y equals negative three and on a mastery check it says, you know, if I had this as an example, if on the master check, it said something like what is the amplitude and you put negative three, that would be wrong because the amplitude is always positive. And what the amplitude is, it is the official definition here. You can read, but basically it is how far you're going from your mountain peaks to the middle. So if I had some middle line here, which is what they're talking about the equilibrium position, it would be right in the middle. So from the middle from the top to the middle or from the middle to the bottom, it's how far that distance is. That is your a okay, the period is, you know what, there's something wrong with this notes. It is two pi over B, I'm going to change this. So you should already have it on your nodes correct. I don't like this word distance. That's kind of gives the wrong feeling. It's really, I don't know what I'm going to put off to think about this for a little bit. It's something about the X values or the change in X values. It's basically something about the domain. How far you go on the X values that's required for the function to complete one full cycle. So if I went back and looked at our example on this, I started with cosine at the top of a peak here, a mountain peak, and then how far did I go on domain values until I got back to a mountain peak? I went two pi. So the cycle of this one, one period for this example, a period would be two pi. Same with this one, the sign started here at zero, went and it's going up, so when does that happen again right here? It's a zero and it's headed up. So at two pi is the period. So whoops, on these examples, the way you've calculate the period is you take the coefficient of X, right there, whatever that coefficient is, you say two pi, and then you divide by that. So normally, the cycle, or the period, excuse me, the period is two by well cycle, same thing. The period is two pi, but if it's, that's only if the B is a one. That's where the parent function. And then frequency is just the reciprocal. Now, why do we need that? Because a lot of times, frequencies are talked about. In fact, this is important. You should write something like this. It is also very similar to a rate of change. And what I mean by that is we're going to see in some of our application problems. You'll see something like the number of cycles, the number of cycles or repetitions per minute or per second. And when you see something like that, that really fast should jump out at you as it's a rate, and that's what frequency is like. It's the number of cycles the graph completes in a given interval or in a given amount of time. All right, that was a lot of explanation on that. Let's get to using this stuff. I do want to show you a quick example of something and point this out. This mastery check, at least for my students, there is no graphing calculator. You really need to be able to do this stuff without a graphing calculator. So as you're going through the practice, a graphing calculator can be used to check your work if you want to check it, but you really have to be able to do this without it. Check with your teacher if you're not sure about that, but it should be no graphing calculator. So I've plugged in Y equals sine of four X if you have a graphic out there, you could grab it real quick and just kind of do this with me or just watch. I'm going to do this real fast. If we do a standard window zoom 6, oh wait, wait, I got to make sure that I am in here, stop that. What's my mode? My mode's degrees. Okay, that's going to throw it off. Let's go to radians. So we're going to graph these things in radians. Always graph it in radians not degrees. All right, so there you can see, I've got these squiggly lines. That's not really a very good window, because really my Y values, I don't need that. Let's make my Y values my minimum. Well, the amplitude in the front here is just a one. So let's just go down to maybe negative three. And positive three, and then look at it. Okay, that's better. And then usually because we're doing in terms of pi, if my window for my X values, maybe if I went negative two and then second, this carrot button right here, that'll give me a pie. See that negative two pi and it enter, and it'll just give you the decimal. Pretty cool stuff. And then again, two pi, so second two, second pi, I'm boom. And then when I graph it, I'm going from negative two pi to positive two pi, and you can see all the cycles going on. So there's the period is pretty short because we have a four here. We're doing two pi divided by four so the period gets much smaller. Okay, with that, now let's graph it without a calculator. Let's see how we do this. Number one number one, what's the amplitude? It is the absolute value of a, so we put a three here. The period is going to be two pi over the value of B in this case, B is a two, so we say that. And then we're going to say pi now the frequency is the reciprocal of this. So we just say one over pi. Okay, let's stop right here. You don't need any of the X intercepts or maximums yet. We're going to write those out in a second, but you don't need to write them out before you graph it. Let's try to make this graphing as easy as possible. So sine is that the origin or a maximum point. Remember what sign does? Sine starts with the origin. Right here in the origin. Now, it's going to repeat, in this case, the period is pi. So I'm coming out here to pi and I'm starting over again. So how does sign start? It goes up right here. So then you go out one period, and you force the graph to go up. Okay? I can not over stress how important that is to go out one period first. Then let's go halfway in between these. So what is sine graph doing halfway in between? It's coming back down with an X intercept. Let me go back to our original function, which was this one right here. If you remember sign, it goes up, down and up. So then if we go to our graph, it's got to go up, down, and then starts back over. And then let's go on here to another pi. We'll go out one more period. And again, it's going to be going up there. So that means in between, it's got to be coming down. So I go in between and it's got to be coming down. Let's go right over here. It's got to be coming down. So now you can get a little bit of a visual. Now let's go halfway between these points. And this is going to be our maximum how high up do we go? One, two, three. Right there. And then halfway here, one, two, three. There we have our minimum. Halfway between these are amplitude is one, two, three. Halfway between these two points, we have our minimum, one, two, three, and then you can continue this pattern over and over and over for as much as your graph will take it. All right, let's fill in the, let's fill in the graph here. I'll speed this up. Okay, there's our graph. Now if we were to write out what all the X intercepts are, this goes on for infinity, both directions left and right. So what I'm going to say is that all of the X intercepts, if you notice how often do these happen for sign, every single pi over two. So it starts at zero, it goes here, and here, and here, so you have to look at the individual graph. I'm just going to list the X intercepts as all the X intercepts are going to happen every N pi over two. This is kind of like what mister breast hot with how many times the co terminal circles go around and around. It's the same type of idea. We say N is an integer. And it happens every pi over two, a second pi over two, a third pi over two, a fourth power two, and so forth. Max and men's are pretty similar. So the max is going to happen right here and here. And here, an over and over and over. So this first one happens at pi over four. So we say the first max is at pi over four, and then it repeats every single period, and the periods of pi, so we just say plus N pi, meaning all the infinite number of pies. And then what's the maximum value, it's a three. So the maximum is a three, there's only one maximum. It's a three, but we're listing every possible maximum coordinate point. That's what this is. And then the minimum coordinate points where's a minimum right here. So what is this? That value right there in between pi over two and pi is going to be three pi over four. And then how often is a minimum repeat? Every cycle. It repeats again on a minimum, so we say plus N pi comma, so there's the X value, and then the Y value is a negative three. Okay, done with number one. On to number two. Number two, we get to do a cosine graph. Let's figure out this amplitude first, so the amplitude is two. Ignore the negative because we do absolute value. Period is two pi over B well, in this case, this case B is one half. So we're going to say one half. Well, how in the world do you do that? You don't divide fractions. You multiply reciprocals. So this means really this means two pi times two. So the period is I'm squeezing this in here for pi. That's the period right there. Four pi. Okay, what else? We can write down the frequency, one over four pi. For pi frequency is a little confusing. Just remember, you'll see when we get to word problems, it's a little bit easier to understand frequency, but it's usually like one over four pi of a circle every single second or minute or something like that. So it's kind of like a rate of change. All right, we don't need to write out X intercepts max mins, anything like that in order to graph it. So really what you need when you're trying to identify everything, you need the amplitude, and you need the period. Once you have those two things, and you know what the cosine and that it's a cosine or sine, you can graph the rest of this. It's really not too bad. So cosine. Does that start at the origin? Does it start with a max? Does it start with a min? What's going on here? Normally cosine is with a maximum. You remember that? It would go up here at one. But because the amplitude is a two, and it's negative in front, we're going to start with a minimum. So this is an example of starting down here with a minimum, because cosine does not go through the origin. All right, what next? Let's use the period. The period is four pi. Wow, four pi. So this is what this means. Cosine means we count all the way out to four pi, which is way off here off the screen. So if we could go all the way off the screen to four pi, then we'd have another minimum way, way out there. So what would be halfway to four pi? Two pi would be. What's halfway if you got after the period, if you have two minimums, halfway there would be a maximum up here at positive two. Okay, let me repeat that. You're here at zero with a minimum. We've got some minimum value way off there at four pi. You've got another minimum value. And so halfway to four pi is a maximum value. And then halfway between the min and the max right here at pi, what are you going to have an X intercept? And then you can fill in the rest of this graph. It's going to kind of go slowly curving there. Slowly curving here. Something like this, and hey, not bad. Tidy that up a little bit. All right. And let's quickly list what all the X intercepts are. So we see here we have an X intercept of pi. So they're going to happen every pi, and then on top of that, the period, how do we how do we list that? Shoot, what is this? We start here at pi, and then we get two pi and then three pi, so it's coming back down at three pi, so we're going to add another two pi to it. Am I doing that right? Two pi, and then an N, because it's going to happen every extra two pi. Yeah, there we go. It's happening here at pi, it's going to happen to get it two pi from here at three pi. It's going to then happen again at 5 pi and 7 pi as it goes up and down, up and down. So there we go. This would be the pattern. You start at pi, and then you add an N number of two pies. There's our X intercepts. Where's the maximum? Maximum happens here at two pi. So we're going to say two pi plus and then it happens again every period of four pi N so I could just say, well, I would just say four pi N comma was the maximum value, it is a two. And then the minimum, the minimums happen. Every zero, and then it happens every four pi every period. So you don't have to write zero plus. I mean, you can. It's not really necessary. So we'll just say every four pi N comma, negative two. That covers every single minimum value on the entire graph. For number three and four, we're not going to worry about all those X intercepts mins and maxes. We had enough practice on that. Let's just focus on the easy stuff amplitude. In this case, it's a two. And then the period period is two pi over B and the B again is one half. That's going to equal four pi. Remember you multiply the reciprocal. And now I've got a vertical shift, this plus one, so that's a little bit different. What I'm going to do is do an imaginary line right here. That imaginary line shifted down just a little bit, is going to be our new middle point, our new equilibrium. It's not actually part of the graph. I just like to draw it there. So you can think of that we've shifted the whole thing up. So this is kind of like our quote unquote new X axis. Now from there, let's start graphing our stuff here. Normally sine is going to be right here in the middle. Yep. So we're going to start off right there in the middle, our new middle our equilibrium point, because it's shifted up one. And then the pi, the period is four pi. Well, what in the world, four pi, there's two pi four pi off the grid. So it's similar to our last problem. So what do we do? Well, remember what sign does, it goes up, down, and back up, and then it's restarting. So halfway, which is two pi is going to be back in the middle. You've got to think sign here, right? Sine goes up, down, up. And so this and this starting point and point middle point. So that's what's going on right there. So that means halfway there, this is where we have our maximum. What's the amplitude, the amplitudes two, so from the middle, we go up two. So really, the coordinate point right there, the corner point is pi comma three, but that's because it started off in the middle at one and shifted up. And then let's go out another pi unit. So how far to pi, one, two, three, four, one, two, three, four. Now we're at three pi, and that will go one, two, down two. And we could go this way, but it's off the grid. So let's fill in the mountain peak and the valley that was pretty sloppy. Oh my goodness. I need help on learning how to draw. And oh, I should have gone down lower. Something like that. There is our sine curve. So obviously you can see, how's the grading on this going to be? Well, you should try to be as accurate as possible, but you can see even beyond pretty sloppy. Now really this red line is not part of the graph. I probably shouldn't have it there to be accurate. But if you lightly sketch a dashed line right there to help you out, I think that's okay to give you a better feel for it. So I'm going to move this over here to number of problem number four. You're going to try number four on your own, but let me help you with something here. And that is three minus four cosine two X. So you've got to make sure you understand this is the same thing as if I took that three and said plus three at the very end. Okay, so it's shifting up three. So I'm going to take that off. And with your line, you're going to go up one, two, and three. And then from there, negative four, so you have a negative cosine graph. See if you can fill out the rest of this. Okay, pretty sloppy stuff. I know going on here, but hopefully you've got something that looks like this. The period was a pie, amplitude was four, and then so when I trying to do this, I start off with the down force. So I went down from the cosine negative. So that means instead of a starting up the top, it's going to go one, two, three, four, start down here. And then I the first thing I did is I repeated every pie. Every period I put another valley. And then another valley every pie. And then halfway between those, that would be where you'd have your mountain peak. Halfway between the valleys is the mountain peak, and then so forth, then I could just kind of start filling in the dots the rest of them. And there's your cosine graph. Okay, the lessons done, I just want to summarize this, so here you go. If this has been confusing at all, watch this part right now. And if you get a little bit confused as you're doing the practice, maybe come back and just watch this very end of the lesson. So let's review something. Cosine graph. In fact, how about this? Let's scratch this off. If it's not different in your notes, I'm changing it right now. I'm going to make up a new one. I'll write these out right now. There, I like these problems better. Okay, let's do this. So you have to identify always when you're graphing these, just real quickly identify what the amplitude is, which in this case is a three, and then you identify the period, you've got to figure out that the period, which is two pi over B in this case, it's two pi over two. That equals pi. Okay, once you have that, then you start with cosine, cosine is starting with the maximum value. So you go one, two, three, or in other words, the value of whatever a is, then you've got to take the period. These are the only two things you need, and boom, you can sketch the rest. So the period we go out one pi right there in the graph is repeating. So if I've got a maximum value here, I have to have a maximum value there. What is halfway between the maximum values? A minimum value down here one, two, three, at negative three. So then what's halfway between the minimum and the maximum, an X intercept, or an equilibrium point, kind of depending on if you've shifted up or down, okay? So then you keep going. That's what you need to do, and then from here, you can kind of see what the pattern is doing every line, you've got an X intercept, a minimum, and so forth. And then on sign graph, sign graph is a little trickier than cosine because of all of the X intercepts. It's so let's remind ourselves. It goes through the origin. Boom, right there in the origin. There's no shifting up and down, so that's pretty easy. Now what's the period? It's two pi. Over B, which in this case is a one, so the period is two pi. And then we go way off here to two pi. Now sine, this is where students get confused. In fact, I've redoing this conclusion because of some students who kept messing this up. Sine is going up right here. Oh no, no, it's not going up. It's going down. So because of the negative in front. So if I'm going down right here, that means when I get to the period, it's got to be going down again. So be very careful, make those little tick marks. Then halfway in between, you've got another X intercept. That's where students mess up. They forget that in the middle is an X intercept. And that one has to be going up. Then from there, you can go in between. And then the amplitude is just the amplitude is just a one. So we go negative one here and positive one here. And then you can start filling in the rest of the graph. There you go for cosine sine, I know this stuff is brand new to you. You never graphed anything like this before, so it can be a bit of a struggle, but if you can get this lesson down, it'll make the second lesson so much easier. Rock that mastery check and I'll see you back in the next lesson.