Calculus: Graphing Trigonometric Functions

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Special Functions
Trigonometric Functions
Graphing Trigonometric Functions Page [1 of 3]
Now, what about the graphs of these functions. How would you actually graph them? Because we'll be looking at graphs of these trigonometric functions, so where do we go from here? Well, the graphs actually - suppose we wanted to graph, let's say the sine function. So F of X equals sine of X, but before we even graph that, I've got to tell you about the x's here.
Traditionally we think about angles measured in degrees, and that's great, and as you remember, one cycle around represents 360 degrees. And we feel very comfortable with that, we know that this right angle is 90 degrees. If you just go one over, and do a straight line that's 180 degrees. But why should once around be called 360 degrees? Sort of an interesting question to think about, right? Why 360 degrees? Maybe it should be a thousand degrees, maybe we should have all started off mathematics, once around is a thousand degrees. So then, half way around would be 500 degrees, and then a right angle would be 250 degrees. Why don't we do it that way? Well, you know what the answer is? It's really pretty funny. Why once around is 360 degrees?
Well because a long time ago, when people were thinking about these trigonometric functions, they were thinking one complete cycle, once around, is like going through a complete calendar year. And they thought that there were about 360 days in the calendar year, which is actually pretty close. And so therefore, they called one cycle around 360 degrees. Isn't that an utterly ludicrous, ridiculous reason to define a mathematical object by how many days there are in the year? That's the craziest, like you know going and getting a psychic reading and figuring out therefore we should call Calculus something else. I mean it's just crazy, but that's what it was, and that's how it stayed, and in fact, we're all very comfortable with that notation, including myself. But for Calculus, Calculus is a science of mathematics, where really we need careful attention to detail. It turns out that if we use that crazy artificial, 360 degrees, things get very complicated. We can make things easier if we actually use what are called radian. So I want to now give you a little reminder, crash course about radian measure of angles.
So it all starts with a much more natural object, let's not look at a calendar, because who cares. Let's instead look at the circle. And if you look at the circle, it's beautiful, the circle is really pretty, really pretty. Now, let's suppose that we have a circle, that's radius one. So the radius of this circle is one. Sort of a standard circle, the radius is one unit. Then what is the length around the whole thing, the perimeter? Or in this case, with circles, we call it the circumference. So what's the circumference of the circle? Well, if you remember the formula, you know it's two pi r. Now here r, the radius is one, so the circumference once around is two pi, where pi is 3.14 so on. So, actually that seems to be a nice number to call one complete thing around. So let's now call, once around, two pi, and we'll call those units, radians. So radians, so once around, once around equals two pi radian. And where am I getting that, from? I'm taking a circle of radius one, I'm fixing that, standard circle, going once around and seeing how much material is required for me to go once around two pi. So I'm calling those radians.
Well, now armed with that basic idea, you can now find the radian measures of a lot of angles. For example, what's half way around? Well, that would be 180 degrees before, so what this by the way, says is in degrees, this is 360 degrees and that would equal two pi radian. Well, it would be half way around. Well, half way around would be 180 degrees, and that would be half of two pi, would be just pi. So, 180 degrees equals pi radian. And just a conversion from one unit to another, it's like metric to feet and yards, and so forth. What do we call that British or English, English? And then what about 90 degrees - well, 90 degrees would be what? Well, that would be half of the 180 so that would be half of pi, so it's pi over two. So pi over two is the radian way of saying 90 degrees, and 45 degrees would be half of that, so pi over four, and you get the idea. Now in fact, there's a way of actually converting from radian to degrees and it follows right from this fact right here. You see, because if 180 degrees equals pi radian, then what's one degree equal? Well, if I divide both sides by 180, on this side, I would see one degree and on this side, I would see pi over 180 radian. So this is a way of converting from degrees to radians. If you have five degrees, you just multiply the five by pi over 180 and you get the radian.
Similar, if you want to convert the other way, what you would do is divide both sides of this by pi, and see that 180 over pi degrees equals one radian. So if you want to know how many degrees is pi over six radian, you would take pi over six and multiply it by 180 over pi and you'd get how many degrees it is. These conversion uh, these conversion facts, these conversion identities, I don't think you should memorize. I think instead you should just remember once around is two pi, and that will then give you all these things, in particular this. And then you can use that to convert, by just dividing by 180 or dividing by pi, whatever is necessary.
So anyway, this is going to be the measure of angles for Calculus, and we're not going to use the degrees. And, and you may say, "Well what if I really like the degrees, can I use them anyway?" The answer is no! Calculus actually won't work that well, if you use the degrees. You really have to get accustomed to using radians. It may take a while, but once you get them, no problem. Anyway, all out of a circle of radius one.
Okay, so now armed with that basic idea, let's see if we can figure out what the graph of this looks like. Okay, what does the graph look like? We want to graph F of X equals sine x, you may remember how this goes. It's a very nice, periodic function that keeps going up and down, in a very almost hypnotic way. And here is what the graph looks like, really pretty, really pretty. So it starts off here at zero, and then it starts to go up very gradually and it gets to the highest point at one, that value there is one, that's one. And then it starts to fall; now that highest point, by the way, is achieved at pi over two, which we used to think of as 90 degrees. This is actually pi over two and then the function starts to fall very gracefully and then at zero, it crosses the x axis at pi, and it keeps coming down here, and this is three pi over two. And then it begins to rise again, and comes back to where we started, makes a complete cycle in two pi, which 360 degrees. You go once around, you make a complete cycle. So that is the sine function, that's the graph of the sine function.
Okay, well now what about the cosine function? Well the cosine function actually is very similar to the sine function. The cosine function is just the sine function, if you just shift it over a little bit. So if we take my, my axis here, and just slide them over a little bit, watch what happens here. Well, it turns out that's going to be the cosine function. So it's just the sine function shifted over by 90 degrees or in our new language, it's just the sine function shifted over by pi over two radian. So the cosine function actually looks like this. And here's the cosine function, and you can see, it's just the sine function, shifted over. So this point here is one, and then it goes down and hits zero at pi over two, which is 90 degrees, and it keeps going down and hits its minimum point at productivity improvement. This is pi here. And that's at minus one, and then it starts to go on the rise, it crosses the x axis. Since the cosine is zero, three pi over two, then it goes back and returns to where it started at this peak of one, at two pi.
These beautiful curves are very related to each other, if you shift one over, you get the other. And the other thing that's interesting is that these curves are actually what's involved in looking at sound. Well, not looking at sound, but listening to sound. The waves that are carried through, for example, the microphones that are pushing through the speakers, through our sound person, actually are all sines and cosines, and in fact, the reason why you hear my voice, is the differentiability in some sense. How we're changing the amplitude and the frequency and in fact if you think of AM radio, that actually stands for amplitude modulation, which means that we're changing how high and how low we go. And FM, which is the real crisp sounding sound, stands for frequency modulation, which means we don't actually deviate from the minus one to one, we don't go up or down. But instead, we put in a lot of waves; we change the amount of frequency of the waves. We wave a lot, rather than going way up and way down. That way we don't go off the band and that's why AM doesn't sound nearly as good as FM. FM sounds great. Anyway, so you can see that sines and cosines are very natural and important to study.
Now, what about the tangent functions? Well, the tangent function actually is one that I do want to tell you a little bit about for a second, because its function is actually much more exotic. Tangent function, I now remind you, looks like this, look at that. Now first of all, what's going on there? You notice first of all its in a lot of pieces, right? There's a piece that goes like this, and then all of the sudden, it jumps down to here. Now I don't know if you can see this line or not, there's a very, very, fine line drawn here. You probably can't see it over the web. But that's actually an asymptote. We'll talk about asymptote in the graphing section later. But that's a line that this tangent function wants to approach, but never actually touches it. It keeps going off to here to infinity and here down to negative infinity.
Why isn't tangent defined there? Well, because none of the tangents is sine over cosine. So if we wanted to see what tangent equals to pi over two, I'd have to plug in, pi over two and for sine and cosine. So I have sine at pi over two over cosine of pi over two. Look at cosine of pi over two; it's zero. So we have something divided by zero, that's undefined. So in reality, we don't cross this line, that's an asymptote. And the tangent function very gracefully starts out at negative infinity and gradually goes up to zero and then continues upward like this to this asymptote and then keeps repeating that process again and again and again. So the tangent graph we can see easily by thinking of it as sine over cosine, so that's pretty neat.
Now, the very last thing I want to tell you about these functions is - well, the values at certain famous points. A lot of times you want to find the values at some big point - let me tell you the values that you should know for certain. You should know - I'll write them first in degrees. You should the trig functions at zero degrees, at 30 degrees, at 45 degrees, at 60 degrees, and at 90 degrees. You should know all the trig value functions just off the top of your head for those values, at least. Now let me convert them to the language of Calculus. Let me convert those to radians. This would be zero radian, and you could try this on your own by just looking at conversion that pi equals 180. This is going to give you pi over six radian. This will give you pi over four radian, like we saw before. This will give you pi over three radian. This will give you pi over two; these are all radian. And if you don't believe me here for example, take pi over three and now multiply that by 180 over pi. If you imagine putting in a 180 over pi here - let's just do that one for fun, so you can see that. I put in a 180 over pi. Well, you see the pi's cancel and three goes into 180, 60 times. So in fact you see 60 degrees.
So, okay now what uh, what are the trig functions for these values? Well, well I'll tell you how a lot of people, in fact how most people, in fact maybe all people uh, remember these things. What they do is they draw, they draw this picture. They draw the XY plane, and they put a unit circle right there in the center. And they start to draw these lines, you know like this. And they read it -- they read it off somehow, and they put things here and then they use the Pythagorean theorem and I think that's great, that's great, if that works for you! But this has never made sense to me personally, so I'm going to show you my own method for remembering these angles because remember I think I've mentioned this already. I don't like memorizing things. So this is the method that I actually invented, believe it or not, when I was in high school because I hated memorizing things, even back then. So here's the method, it's the world's I think, shortest, and in my opinion, easiest way of remembering all the trig functions. So first of all, what do I do? I'm going to put the angles here, and then I'll put sine here, and I'll put cosine here. And the first observation is all I need is sine and cosine. If I know sine and cosine, I know everything. You want tangent, it's sine divided by cosine. You want cosecant; it's one over sine. You want secant; it's one over cosine. You want cotangent, it's cosine over sine. So if you just know these guys, then you're all set.
Okay, so let's make a little chart here. What I'm going to do is I'm going to put in those famous angles we want -- so zero, pi over six, pi over four, pi over three, and pi over two. Those are the angles in radian, in increasing order. I'm going to build a chart right here for you, live. Live on the - you get to watch me. It's sort of fun, here the information superhighway, is working so hard to crunch and share with you all these gigabytes or megabytes or maxi bytes, just so you can watch me draw these lines. Sort of wasteful, but who cares. Anyway, there's the chart and now here's how you fill it in, I think this is really sort of fun.
So what you do is, what you do is, you remember just one basic slogan. The sine is good, the cosine is bad. So the sine is sort of the good, the good sort of bee and the cosine is sort of the dark sheep of the family. Well, step one is to first of all, divide everything through by two, just everywhere on this table, just write divided by two. Don't think about it; just divide everything through by two. So that's pretty easy, just divide everything through by two, not a problem. And then to find the values, all you do is just count, what could be easier than counting? I want you to count. I want you to start up here at zero, and I want you to start counting though at zero -- zero one, two, and count that way and the only rule is just count under a square root. So first, write the square root symbol and then count. So, here I'd write square root of zero. Here I'd write square root of one. Here I'd write square root of two. Here I'd write square root of three. Here I'd write square root of four; what could be easier. Here I did the same thing, but remember cosine is that one. It's that rebel without a cause, so what's I'm going to do with cosine is count, but I'm going to start counting backwards, because cosine is trying to be sort of cool. So square root of zero, square root of one, square root of two, square root of three, square root of four, and that's it. Believe or not, that is it. Because if you notice, the square root of zero is zero over two, that's a zero. This is just square root of one, which is one, so this is one-half. This is the square root of two over two. This is the square root of three over two, and the square root of four, of course is two and two divided by two is one. So there's all the trig values, for sine, at these different angles, and similarly if you work backwards here, you'd see the same kind of thing. In fact, if you're really good, what I've been doing, since I've been doing this you know for about, you know, 60 years, since I was a kid. What I do is I just memorize the sine thing and realize that cosine is to read it backwards. So in fact when I'm doing calculations, I just thinking this one column and thinking down and then remember its backwards and forwards. Anyway, that's my warped little way of remembering the basic trig functions at these points. Um, probably it's better for you to actually use the little circle method, if you understand that, and maybe one day you can type in a message to me on this thing and explain it to me. I'd love to learn about it. But until then, I'm sticking with this old thing that I invented back when I was in high school. That's just sort of a basic overview of trig business and now what I want to do is take a look at the Calculus of the trig functions. So I'll see you over at the Calculus section of trig. Okay, bye.