Why hyperbolic functions are actually really nice

this video is all about hyperbolic functions so often students are given these kind of bizarre Expressions but why are they connected geometrically to hyperbolas that are so useful and if you're a fan of the channel you will see hyperbolic functions coming up in my videos about bubbles about the math of War and the geometry of hyperbolic spaces more generally let's start on the analytic side with these messy formulas if I start with a function like e to the x I can flip it over the y- AIS by considering e to the Min - x and then I can flip that over the x-axis by considering the negative of e to the negative of X but check out what happens with just good old cosine of x cosine of x is what we call an even function and that means that when I flip it over the Y AIS by considering cosine of minus X then I get back exactly the same thing now I'm going to connect this to hyperbolic functions in just a moment don't you worry but I want to in general Define what even and odd functions are an even function is one where f of negative X is the same thing as F ofx and an odd function is one where F of negx is the negative of f ofx then there's something kind of cool I can do with even an odd functions if you give me an F ofx then I can Define what I will call F even and F odd F even is just like the average of F and F negx and F odd is like the average between F and negative f ofx and I haven't made up this naming these actually are even and odd functions like if I take F even here and I put negatives in well negative of a negative is just the same thing of X and this is exactly where I started so this really is even in contrast with f odd if I was to put negatives in everywhere the negative the negative cleans up when I flip this around it's now got a negative stuck out the front so F odd is indeed an odd function but what's cool about F even and F odd is that I can take any function f ofx and write it as the sum of an even component and an odd component indeed those f of negative XS they cancel you get two copies of f ofx divid two if I substitute this in this is just F ofx okay all of that Preamble was for this particular moment let me plug in this specific function e to the X as my function in this case my f even is E X Plus e- X over2 and my odd e xus e- x / 2 these are even and odd functions and the sum of them is just e to X I am going to call these hyperbolic cosine and hyperbolic sign so from a analytical perspective just looking at the formulas the definition of hyperbolic cosine and hyperbolic sign is just the split into its even and its odd component I haven't yet explained why this has anything to do with hyperbolas but I'm just saying that hyperbolic cosine hyperbolic sign are natural things to consider they're the even and the odd components of exponential a function that we know has enormous importance in mathematics now that I've defined them I can graph them to graph hyperbolic cosine of x I'm just going to take the e to the X I'm going to take the E Theus X and I'm going to take their average which gives this very nice plot of hyperbolic cosine and indeed visually that looks like an even function and then hyperbolic sign is going to just be the average of e to the X and netive E to thex which creates this odd function and then you can check that if you add up hyperbolic sign and hyperbolic cosine at any different point it's going to add up to what you would expect for e to the X visually if you know about Taylor series and calculus I have one more cool thing to tell you or skip forward to the next time stramp for the geometry this is the tailor series for exponential 1 + x + x^2 2 factorial and so forth if this was a polom the even powers of that polinomial are going to behave like an even function that's where the name comes and the odd powers like an odd function so I can take this Taylor series expansion for Ed the X and I can imagine breaking it up all of the even terms that goes into the even component of exponential in other words hyperbolic cosine and all of the odd terms become hyperbolic sign so you get power series definitions for hyperbolic Co and hyperbolic sign out of this even odd analysis for the tailor series of e to the X and I can either use the series representation or my original definition just to check that you have relationships like that the derivative of hyperbolic cosine is just hyperbolic sign indeed you can check that for every term when you take a derivative it brings the power down cancels one of the things in factorial and gives you exactly what you get for hyperbolic sign similarly the derivative of hyperbolic sign is hyperbolic cosine except for a missing minus sign this is very similar to the derivative relationship between s and cosine the non-hyperbolic versions now we need to connect this to the geometry here I put up the plots of a circle and of a hyperbola and they're very similar for the circle it's just x^2 + y^2 = 1 and for the hyperbola well x^2 - y^2 = to 1 now you might remember the Pythagorean identity for trigonometry that sin^2 + co^ S equal to 1 and satisfies the equation of a circle let's do the same kind of analysis for the hyperbola for this equation x^2 - y equal to 1 if I take X to be hyperbolic cosine of some other variable let's call it Theta and Y to be hyperbolic s then I can just go and take this X and Y and I can plug them into the definition of my hyperbola well if I expand all of these out I get this sort of Long messy expression there's a whole lot of cancelling going on and what do I get it's just equal to one and this tells me that hyperbolic cosine and hyperbolic sign they satisfy the equation equation of a hyperbola that's a little bit satisfying you might think I made my big connection except actually lots of pairs of functions like that satisfy the equation of a hyperbola like xal tan Theta and Y is secant Theta you can plug it in and check it yourself that also when you plug it in is equal to one so it's nice that hyperbolic cosine hyperbolic sign do satisfy the equation of a hyperbola but there's many ways to travel a hyperbola parametrically and this is only only one of them so why is this one so special to more deeply understand this connection let's go back to the definition of the regular trig functions the way I Define s of theta and COS of theta is I imagine that I am at a point on the unit circle making some angle Theta with positive horizontal axis and then I Define the horizontal component of that point to be cosine of theta and I Define the vertical component to be s of theta these are the geometric definitions of the COS and sin function then I can deduce the graph of say cosine of theta where I'm just going to follow along with what my horizontal component is doing and let the height of this cosine function just be the length of that horizontal component this gives the familiar cosine graph and similarly we can generate a plot of s of theta by looking at how the vertical component of my point is moving as I increase Theta now that standard definition of cosine and S that's related to this angle of theta could actually be restated in terms of area that is I can imagine that at this angle of theta I also have a region that's shaded out and I can ask what's the area of that region this is a simple computation to do theta can go between 0 and 2 pi so if you're at an angle of theta you're a proportion th over 2 pi of the total area of a circle which for a unit circle isk * 1^ 2 so all this is to say is that the area of that pizza slice is equal to the angle divided by two to get rid of this division by two I'm actually going to set a larger area as my definition of the area here so it's going both above and beneath the x-axis by that amount Theta and with this larger definition of area I have the formula that area equals angle what I can do then is Imagine in my definition of cosine and S of theta is that I instead replace it with the area divided by two I imagine this is half of that total area and then my definition of cosine and S are just related to the area it says you take the point on the circle that has this particular area and the horizontal component of that point is cosine and the vertical component is s that is to say I'm sort of redefining cosine and S to be functions depending on the area enclosed by wherever your point is as opposed to the angle Theta either is perfectly fine for trigonometry but I'm doing all of this so I can do the same area definition to define geometrically the hyperbolic functions let me put a point on the hyperbola and let me imagine that I have an area enclosed that same area divided by two if you wanted to do the portion underneath the axis it would be a total area of a so I'm asking where is the point such that you're en closing this particular area and then I want to look at the horizontal and vertical components of that point that I will geometrically Define to be hyperbolic cosine and hyperbolic sign if you give me an area then that area determines a point and I can give you back hyperbolic cosine and hyperbolic s with this geometric definition in mind we can generate the same plots for hyperbolic functions that we saw before for hyperbolic sign of a then the height of that function at any point is just the height of the points on the hyperbola and for hyperbolic coine then the height of the function is just the horizontal components of the points on the hyperbola these give these beautiful functions for hyperbolic cosine and S so as of right now I have two different definitions of hyperbolic Co and sign floating around I have my analytic definition as the even and odd components of the function e to X and I have my geometric definition as the horizontal and vertical components of the point on a hyper hyperbola that is swept out a specific area so are they the same I'd shown that the exponential ones I could plug in in could satisfy the equation as did many other functions but do they have this nice area property to help me do that I need to get a handle on what this area really is and I'm going to note that I can add in this second region B and A and B together just for form a right triangle so if I want to figure out what the area of a is it's the area of the triangle and an area of a triangle is 1/2 base times height so in this case so in this case it's hyperbolic cosine and hyperbolic S as my base and my height of my triangle and then I subtract off the region B I'm doing this because it's going to be a little easier to compute out the area of B and then take the difference than it is to go for a directly so let's go for that area of B now the calculus students among you will find this a love practice of integration which is you know how we figure out areas in calculus I noticed my curve is x^2 - y^2 1 that was the definition of a hyperbola I can rephrase that and say that Y is square otk of x^2 minus 1 and so if I imagine slicing this region of B up into a whole bunch of vertical strips that look like this pink thing then the vertical strips are going from Z up to a height of < TK of x^2 - 1 and thus I'm going to take take an intergral of such things I'm going to add up all of these little vertical strips here I'm going to add up all of these x squ minus 1es and I'm doing it from a left point of one all the way out to a right point of hyperbolic cosine this is just an integral I'll let you figure out the details of evaluating that integral down in the comments if you so interested but here it is there's the mess and for our purposes what I really want to focus in is that because I'm evaluating this at hyperbolic cosine it turns out a lot of things simplify like look at these two expressions where I've got < TK of x^2 minus 1 if I plug the bottom of one those are just going to be zero so I don't have to worry about those at all same thing logarithm of one is zero when I plug in one everything's going to be zero go away but when I plug in hyperbolic cosine well hyperbolic cosine squ minus one is just hyperbolic s squar we've already proven this identity square root that and you get hyperbolic sign so you plug hyperbolic cosine into either of those yellow expressions and you just get hyperbolic sign okay so what does that leave us with leaves us with this expression here where I've plugged in the hyperbolic Co and S but now I can leverage my original definition we defined hyperbolic Co and hyperbolic sign as the even and odd components of exponential function their sum is just the exponential and then it's easy I have logarithm of an exponential this is just going to be the value of a and well divided by two so subtracting a divided by two I got the same expression on the front of both of these which I can cancel and as a result the area of a is nothing but a / 2 so what I've shown is that with hyperbolic s and cosine defined as the even and odd part of exponential that was crucial to my derivation I then get that geometric property that I wanted such that the area enclosed was just going to be a / two so these two different definitions are in fact the same now I want to just scratch the surface on one other aspect of all of this where we talk about these as complex functions you might be familiar with Oilers famous identity that e to the iix is cos x plus I sin of X contrast that famous identity with our definition of the hyperbolic functions as the sum of hyperbolic cosine and hyperbolic sign in our presentation this is a definition not a theorem but notice the similarities between these if I just plugged in i instead everywhere well then I have e to thex on both sides and so then if I compare these cosine the real component is nothing but hyperbolic cosine of I * X and similarly s of X is hyperbolic s of I * X ided up by I and so really cosine and hyperbolic cosine s and hyperbolic sign can just be thought of as different sides of the same larger coin now I'm a math YouTuber I love math videos and I hope that you do too but to really Master mathematics you have to actually get your hands dirty doing mathematics and that's why I am such a fan of the sponsor of today's video which is brilliant.org brilliant is an online learning platform with thousands of courses that are just delightfully interactive you get to play with the animations and visualize what's going on and you get to test yourself on your understanding as you learn the content I had fun the other day going down the rabbit hole of beautiful geometry course which anyone who enjoyed this video might really like to play around with so if you're looking to improve your mathematics skills or you just want to see a bunch of cool mathematics I really encourage you to check out brilliant.org Trevor bazit that's me the link is down in the description to check out everything that they have for free for full 30 days or the first 200 of you who click the link are going to get an additional 20% off an annual premium subscription with that said I hope you enjoyed this video if you have any questions leave them down in the comments below and we'll do some more math in the next video

Channel: Dr. Trefor Bazett

Views: 122,617

Keywords: Solution, Example, math, hyperbolic functions, engineering mathematics, calculus, hyperbola, hyperbolic geometry, hyperbolic trig functions, definition, inverse hyperbolic functions, derivative of hyperbolic functions, euler's identity, area enclosed, hyperbolic trig

Id: HnHnEnkZpJA

Length: 16min 3sec (963 seconds)

Published: Mon Oct 30 2023