 Welcome back to our lecture series Math 12-10, Calculus I for students at Southern Utah University. As usual, I'll be your professor today, Dr. Andrew Misseldine. As we come to the close of chapter three, I want to introduce one more family of functions, the so-called hyperbolic functions. Now the hyperbolic functions are very similar to the six trigonometric functions we've studied previously. And because of that, their names are going to mimic those of the six trigonometric functions. Sometimes to distinguish between these, we call the usual trigonometric functions, sine, cosine, tan, et cetera. We might call them the circular trigonometric functions or just the trigonometric functions for short. These six functions we would call the hyperbolic trigonometric functions or just hyperbolic functions for short. And so there are going to be six hyperbolic functions and their names are defined as following. You first get the hyperbolic sine function, which is denoted, it's just abbreviated as s i in h of x. It's sometimes called cinch for short, the hyperbolic sine function. It is defined as e to the x minus e to the negative x over two. Similarly, hyperbolic cosine, which is denoted c o s h of x are called cauch for short. Cauch of x is defined to be e to the x plus e to the negative x over two. And so you can see these two functions cinch and cauch are defined very similar using the natural exponential e to the x. The only difference in their formula is cinch has a negative sign right here and cauch has a positive sign right here. Now, it might not be obvious why anyone would call these things trigonometric in any regard. What does this function have to do with sine and what does this function have to do with cosine? Well, if I dare say that these functions cinch and cauch are the imaginary friends of sine and cosine. And I mean that, I mean, it's a horrible pun, mind you, but I mean that in the complex number sense, that with the real numbers, which in this calculus series will focus on real numbers and no imaginary numbers whatsoever, in the complex number sense, sine and cosine behave very similar to cauch and cinch if we reflect them across the imaginary axis essentially. If you take, for example, cosine of ix, so if you take cosine evaluated imaginary angle, this is actually equal to cauch of x. So there is some imaginary relationships between cautions and cosine, same thing with cinch and sine, whether we won't delve into it this moment, just take these as two new functions cinch and cauch. Then we're going to define the four other hyperbolic functions by analogous relationships like we saw with circular trigonometry. So hyperbolic tangent, I guess you could call it taunch or something if you want to, that's defined to be cinch divided by cauch, that is sine divided by cosine in the hyperbolic sense. Hyperbolic cotangent, I suppose you could call it cough, that's just going to be the reciprocal of hyperbolic tangent, it's cauch divided by cinch. Hyperbolic secant, this is just the reciprocal of cauch, and then hyperbolic cosecant, there's not even a vowel to play with that one, that's what I'll call it, that's the reciprocal of cinch. And so you get the six hyperbolic functions in that regard. So the hyperbolic functions get their names, again because of this imaginary relationship we see before, but even if you don't use the imaginary relationship, it turns out that we can still find a geometric analog between the hyperbolic functions and the trigonometric functions. Because in fact, despite the definitions of hyperbolic sine and cosine, the definitions of the hyperbolic functions are analogous to the trigonometric ones. Well, you see that for these four, but why do cinch and cauch resemble sine and cosine in any way? Well, the only real difference it turns out is from the Pythagorean equation. Now for the circular trigonometric functions, you know the relationship that cosine squared plus sine squared is equal to one. This is derived from the unit circle, the unit circle. On the other hand, the hyperbolic functions gets their name because they satisfy a hyperbolic relationship. You get that cauch squared minus cinch squared is equal to one, which the proving this hyperbolic identity is fairly straightforward. If you take cauch squared minus sine squared, you can plug in their definitions, e to the x plus e to the negative x over two squared minus e to the x minus e to the negative x over two squared. If you foil that thing out, you're going to have to, well, you square the two, you're going to get a one fourth. So there's a one fourth over both of these, which we factor out to here. And so we're going to have to foil out the e to the x plus e to the negative x squared. You'll have to do that twice, right? So when you do that, you're going to get e to the two x plus two plus e to the negative two x. You get that twice, but there's a negative sign. So things are going to cancel out. These guys will cancel out. These guys will cancel out. And so you're going to get two minus a negative two, which is equal to four. So you get four over four, which is equal to one. So we see that the hyperbolic function satisfy this relationship right here. But this, when coming back to the Pythagorean equation for circular trigonometry, you get cosine squared plus sine squared equals one. This is just taking a parameterization of the unit circle x squared plus y squared equals one. On the other hand, the hyperbolic functions just give us a parameterization of the unit hyperbola x squared minus y squared is equal to one. So let's use Desmos to compare the differences between the circular trigonometric functions that we're used to and these new hyperbolic trigonometric functions. So what you see now on the screen is the unit circle drawn in the xy plane. And what we're probably used to is the fact that if you take a various line segment here inside the circle and you consider the angle that's formed between the x axis and this line right here, this angle theta, the coordinates of this point on the unit circle coincide with cosine and sine of that angle, right? So if we take, for example, this angle theta right here, then it turns out this number, 0.697, this is just cosine of that angle. And this other number right here, 0.717, this is just sine of this angle. So the coordinates on the unit circle, which are in correspondence to specific angles with the positive x axis, give us these trigonometric coordinates of cosine and sine. And so I allow the angle to change this cosine and sine of the corresponding angles also changes. So again, as we move this around the unit circle, we get these different cosines and sines and they always coincide with the angle the line forms with the positive x axis. Let's contrast that to the unit hyperbola, which you now see here on the screen. I'm going to zoom out a little bit. Unlike the unit circle, the unit hyperbola is much more, it's unbounded in its region it covers here on the plane. This is given by the equation x squared minus y squared equals one. And the same is also true as we allow this line segment to vary. We get these different hyperbolic coordinates. In which case, I want to emphasize to you what's going on. The angle forms between the positive x axis and this line segment, let's call it theta. Then you see this point has these coordinates, this first coordinate right here, 3.762. This is the hyperbolic cosine of that angle. Then likewise, this number right here, 3.627, this is just the hyperbolic sine of that same angle theta. Therefore, as the angle is allowed to vary, the angle we're forming with the positive x axis, that is, we get these various different coordinates. But the x-coordinate will always be represented by hyperbolic cosine or cosh, and the y-coordinate will always be represented by hyperbolic sine or cinch, just in the same way that trigonometry represents the coordinates of the unit circle in the usual sense. Let's also use decimals to look at the graphs of the six hyperbolic functions. We'll start off with cinch. So you can now see cinch here on the screen, illustrated in orange, and maybe your first take on it is that the graph of cinch doesn't look anything like sine. It looks more like an odd monomial function. You'll notice that this increasing behavior, it's symmetric with respect to the origin. It actually passes through the origin. It looks a lot like an odd monomial, like y equals x cubed or y equals x to the fourth. Like sine, it does pass through the origin. So notice cinch of zero is equal to zero, sine of zero is also zero. The sine function's an odd function, but that's about the only similarities between the graph of cinch and the graph of sine. When we zoom out, again, like I said, this looks a lot like an odd monomial function, but unlike x cubed or x to the fourth, the growth of this function is exponential. So as x approaches infinity, this graph basically looks like one half e to the x. So then you have exponential growth over there. And likewise, when you go over here on the left-hand side, it looks like a reflected exponential graph going towards negative infinity. So as x approaches infinity, a sensual approach infinity, as x approaches negative infinity, sensual approach negative infinity as well. So we see that the domain of cinch is all real numbers, and the range of cinch is likewise all real numbers. Let's take a look now at the graph of Cauch. Cauch looks a lot like an even monomial function, kind of like x squared or x to the fourth or x to the sixth. It really doesn't look like cosine whatsoever. There are some important similarities between cosine and Cauch. For example, you'll notice that Cauch at zero is equal to one, Cauch at zero is equal to one. That's also true for cosine. Cosine at zero is equal to one. Cosine's an even function. Cauch is likewise an even function. The graph is symmetric with respect to the y-axis. But just like cinch and sine, that's about as similar as Cauch and cosine are. They're very different otherwise. Zooming out a little bit, you'll notice that as x approaches infinity, Cauch will go off towards infinity as well. It has exponential growth as you're going off towards infinity. As x approaches infinity, Cauch will be asymptotically the same thing as one-half e to the x. The left-hand side is a reflected exponential function as well. At least its behavior is asymptotically the same thing as that. It kind of looks like, in this case, one-half e to the negative x. Again, asymptotically, those two will be the same. So we have this exponential decay and exponential growth. As x approaches negative infinity, Cauch will approach infinity. We can see that with Cauch, its domain is going to be all real numbers. There's no number for which Cauch is undefined. Cinch and Cauch are not periodic functions like sine and cosine. They're very different in that regard. And then the last thing I want to mention is that the range of Cauch is limited. The range of Cauch is going to be one to infinity. We don't get anything lower than one unless the graph is transformed. Let's take a look now at the hyperbolic tangent function, which when you look at this, the graph actually looks a lot like arctangent, but those similarities are more coincidental than anything else. Why do I say it kind of looks like arctangent? Well, some things to note here is it does pass through the origin. Hyperbolic tangent at zero is equal to zero, just like tangent at zero is equal to zero, which arctangent also does that. It's an odd function. It's symmetric through the origin, but kind of like arctangent, hyperbolic tangent has these asymptotes. But the asymptotes aren't going to be at pi halves or at negative pi halves. They're going to be at one and negative one, which you can see those illustrated here on the screen right now. We have these horizontal asymptotes. So as x approaches infinity, hyperbolic tangent will approach one. And when x approaches negative infinity, hyperbolic tangent will approach negative one. So you see this behavior here. And this comes from the fact that hyperbolic sine and hyperbolic cosine as they go off towards infinity, they're growing at the same rate. Therefore, the ratio, which is hyperbolic tangent, will be one. When you're approaching negative infinity, ascension, cauchy, their behavior, their growth is basically the same, but they're off by a negative sign. So therefore, the ratio will go off to negative one. Notice that the domain of hyperbolic tangent is going to be all real numbers, and its range is going to be from negative one to one, where negative one and one are not included in that. The range doesn't actually touch one or negative one. We can get asymptotically close to it. And that'll do it for hyperbolic tangent. Let's take a look at hyperbolic secant. This is sort of like a curious little function. You look at this one, it looks like a little bump function. When you're near the y-axis, you see this bump. But when you're away from the y-axis, you hardly see anything at all. We'll talk about that just in a second. The thing to remember about hyperbolic secant is just like this traditional circular secant. Hyperbolic secant is the reciprocal function to cosh. So if we put cosh here on the screen, we see that as cosh gets big, hyperbolic secant gets small. And as cosh gets close to one, hyperbolic secant will also get close to one. But from the other side, we see this reciprocity going between cosh and hyperbolic secant. Now, since cosh goes off towards infinity, as x approaches infinity or negative infinity, we see that hyperbolic secant will approach zero as x goes to infinity or as x goes towards negative infinity. So hyperbolic secant actually has a horizontal asymptote at the x-axis. So we see that the domain of hyperbolic secant is all real numbers. This is in contrast, of course, to the circular secant, which is undefined whenever cosine goes to zero. Tension had that same problem. The domain for hyperbolic secant, just like hyperbolic tangent, is all real numbers. But the range of hyperbolic secant is going to be all numbers between zero and one, where in this case, one is included in the range, but zero is not included in the range. Hyperbolic secant is never equal to zero, but it can get asymptotically close to zero. And that'll do it for hyperbolic secant. Let's take a look at hyperbolic cosh secant. This one looks really bizarre compared to hyperbolic secant. When you take the circular secant and circular cosh secant, they look very similar, but there's offset by just a phase change of some kind, it feels like. But when it comes to hyperbolic secant and hyperbolic cosh secant, they're not periodic functions. None of the hyperbolic functions are periodic. But also, we see that the graphs don't resemble each other whatsoever. When you look at hyperbolic cosh secant, it looks kind of like the rational function 1 over x or 1 over x cubed. It's not exactly the same. It's not, it doesn't have that hyperbolic shape, but it kind of does look like that. Some things to notice is that there's going to be some asymptotic behavior. We see that there will be a horizontal asymptote at the x-axis as x approaches infinity, or as x approaches negative infinity, hyperbolic cosh secant will approach zero. Although as x approaches infinity, you'll approach zero from above. And as x approaches negative infinity, you'll approach zero from below. So that's important to note. This graph also has a vertical asymptote. We haven't seen this on any of the other hyperbolic functions, although this is commonplace for the circular trigonometric functions. We see that it has a vertical asymptote at x equals zero. So the domain of hyperbolic cosh secant is all real numbers except for zero. And we also see that the range of hyperbolic cosh secant is all real numbers except for zero. The thing to remember about hyperbolic cosh secant is it's the reciprocal function of cinch. So when cinch gets close to zero, like it passes through the origin, we see that hyperbolic cosh secant actually explodes towards positive infinity or negative infinity. So as x approaches zero from the right, hyperbolic cosh secant will approach positive infinity. But as x approaches zero from the left, hyperbolic cosh secant will approach negative infinity. And so this is the reciprocal behavior passing through the origin. Then as cinch goes off towards infinity, you're going to see that hyperbolic cosh secant here goes off towards zero. And you also see the reciprocal behavior here as x, or as hyperbolic sign goes towards negative infinity. We see that hyperbolic cosh secant is going to go off towards zero as well. You have the horizontal asymptote. And so that's going to give us the graph of hyperbolic cosh secant. So let's look at the last one. Hyperbolic cotangents are koth, I guess, if you want to give cute little names to these things. Koth is sort of a curious location. It kind of looks like it's rational function again, but you'll notice that there's this huge gap happening between the two functions. Why is that? Well, as the reciprocal of toth, tanth, ton, what would you call that, hyperbolic tangent, you see it's the reciprocal going on right here. As hyperbolic tangent approaches its x intercept at the origin, we see that hyperbolic cotangent is going to do the opposite, the reciprocal behavior. So in fact, we see there's a vertical asymptote at the y axis. So as x approaches zero from the right, hyperbolic cotangent will go towards infinity. And as x approaches zero from the left, hyperbolic cotangent will go towards negative infinity. But we also see some vertical, excuse me, horizontal asymptotes on the graph of hyperbolic cotangent. You have an asymptote at y equals one and an asymptote at y equals negative one. Why is that? Well, for hyperbolic tangent, as x goes towards infinity, this thing is going to go off towards one. If you reciprocate that, that is you take the reciprocal of all of those fractions as hyperbolic tangent is going from zero to one, hyperbolic cotangent will do the opposite. It'll go from infinity going to one. So they both approach one, but hyperbolic cotangent will approach one from above, hyperbolic, excuse me, hyperbolic cotangent will approach it from above, hyperbolic tangent will approach one from below. As we approach the other horizontal asymptote at negative one, you see the opposite behavior. As hyperbolic cotangent goes from zero to infinity, we'll approach negative one from above. But hyperbolic cotangent will go from negative infinity to negative one from below. So you can see this reciprocal behavior going on right there. And so the last thing to conclude here is that the domain of hyperbolic cotangent will be all real numbers except for zero. So that's just like what we saw for cosecant. But unlike hyperbolic cosecant, the range is much smaller. The range of hyperbolic cotangent is going to be all, it's going to be all numbers from negative infinity to negative one. And we're also going to get all numbers from one to infinity. So you're going to get negative infinity to negative one union one to infinity, which I should note that one and negative one are not included inside of those ranges because hyperbolic cotangent will never obtain one nor negative one. So from an algebraic sense, the real difference between the hyperbolic functions and the trigonometric functions are their Pythagorean identity. Cosine squared plus sine squared equals one for circular trig. And you get cos squared minus since squared equals one for hyperbolic trig. And so let's look at sort of like the most elemental of all of the trigonometric identities and see how they relate to hyperbolic functions. Well, like we saw with the graphs, sinh is an odd function. So sinh of negative x is equal to negative sinh of x. That's true for sine as well. Cosh is an even function. So you get cosh of negative x is equal to cosh of x. You have the Pythagorean identity. The main difference, of course, is that this is now a minus sign instead of a plus sign. But from this, we can derive other identities. Like if we divide both sides of the equation by cosh squared, you'll get that cosh squared divided by cosh is squared as a one. Sinh squared divided by cosh squared is a hyperbolic tangent squared. And one divided by cosh squared is a hyperbolic secant squared. So we can derive basically a similar Pythagorean identity from this mother identity right here. When it comes to hyperbolic or to circular functions, one plus tangent squared is equal to secant squared. In the hyperbolic realm, it's now a negative sign. We can derive other identities as well. What about when it comes to sums of angles? If you take sinh of x plus y, this equals sinh of x cosh of y plus cosh of x sinh of y. This is the exact same identity when it comes to sine. Sinh of x plus y is equal to sinh of x cosh of y plus cosh of x sinh of y. On the other hand, when you take the angle sum identity for cosh, cosh of x plus y equals cosh of x times cosh of y plus sinh of x times sinh of y. This is a critical difference. When you take cosh on the other hand, cosh of x plus y, this is equal to cosh x cosh y minus sinh of x sinh of y. And so this is the main difference from an algebraic point of view between the hyperbolic functions and the circular functions. When you take the identities of the usual trigonometric functions, those are identical identities for the hyperbolic functions, although on occasion, you switch pluses to minuses and minuses to plus. When you look at it from a naive perspective, the change of plus and minus seems random and arbitrary. It all can be derived basically from this hyperbolic change, this hyperbolic relationship instead of the circular relationship. But with that said, working with hyperbolic functions in many ways is the same thing as working with the circular functions. Because of that, there's sometimes not a strong need to use hyperbolic functions because they behave so much like circular functions. But there are some important differences I want to make mention. So while they could be similar, there are some applications of hyperbolic functions that offer new things. Applications of hyperbolic functions to science and engineering occur when we talk about light that can be used in examples of velocity, electricity, radioactivity, where perhaps the radiation is gradually absorbed or extinguished over time. For the decay can be represented using hyperbolic functions. Hyperbolic functions can be used to model the shape of hanging wires, like if you have, for example, two telephone poles and there's a wire dangling between those. This type of curve can be modeled using a Cauch function. This can also be used to represent velocity waves, waves in the ocean, and other things as well. So the trigonometric functions, the usual circular trigonometric functions are very useful in modeling harmonic motion like oscillations of some kind. We can use these non-oscillating hyperbolic functions to capture other things that have some very important uses in the sciences.