@@DrTrefor I first stumbled on the term, when my textbook used the Gateway Arch in St Louis as an "example" of a parabola, with a fine print note that it's really a hyperbolic cosine. Playing with my graphing calculator, I attempted every combination of hyperbolas and cosines I could think of, like 1/cos(x) and cos(1/x), and couldn't find anything resembling it. Eventually learning it for real, I figured out on my own what properties hyperbolics have in common with standard trig, and could connect the dots on at least that part of its namesake.
@@DrTrefor I think it's the fact that we never learn the area approach you showed in terms of relating cos and sin to the unit circle, which makes it unnatural to think of the hyperbolic parametrizations cosh and sinh in the same way.
I always thought hyperbolic functions were just some weird made up versions of regular trig functions. I didn't realize how intuitive and natural they are.
In some regards, the hyperbolic functions are more natural than the circular functions ('circular' is a more appropriate adjective to use than 'trigonometric').
I've somehow managed to never have a class on hyperbolic functions even though they show up occasionally. This video is mind blowing and really puts together so many disparate puzzle pieces for me. Truly incredible work!
I also made a video on the subject but with detailed computations of the integral. Check the video on my channel for more details: video title: "he Geometric Definition of the Hyperbolic Functions, and Derivation of their Formulas"
Cool integration trick somebody taught me: if you’re integrating some gnarly function over some interval that symmetrically straddles zero (say, between -1 and +1), split the integrand into even and odd functions and see if the even function is more amenable to analysis. This is because the contributions of the odd function will cancel out and can be ignored. EDITED TO CORRECT ERROR THAT WAS POINTED OUT.
Thank you for this video. Hyperbolic trigo is not even taught in schools where I live so most people don’t even know they exist even until they graduate from high school. The complex relation between the regular and hyperbolic trigo functions also explains the similarity between their derivative properties, and their taylor series. The taylor series for sinx and cosx have that alternating factor. The hyperbolic functions have the exact same terms just without the alternating.
That's really, really awesome. I was wondering recently why these functions were called "hyperbolic". The analogy with circle and sin and cos is great!
WOW my mind was blown just in the first few minutes, seeing the beautifully elegant explanation of splitting e^x into an even and odd part, and then it just continues getting better and better 🤯 I've heard plenty of explanations of sinh and cosh but none like this. The other videos on sinh + cosh don't give nearly as much intuitive explanation - just a bunch of symbol mashing and head scratching - so this is much more satisfying.
This is such a clear explanation of hyperbolic functions! What a perfect timing too, since I was wondering about them after my multivariable calculus professor briefly mentioned them in lecture a few days back, but never bothered to go over them in detail since they were irrelevant.
Wow, they just slammed hyperbolic functions on our faces 3 days before calc 1 exam and never heard of them anymore until i was studying special relativity and complex analysis. And even then nobody even bothered explaining them. Glad you did, thank you very much, the passion you put in your videos is tangible
The way i first found hyperbolics was when i was curious on what cos(ix) was, so i used the maclaurin expansion and found it wasthis cool, and surprisingly real valued, mix of e^x and e^-x. It was only much later when i realised that was infact cosh(x). I love hyperbolic functions man
4:40 in to the video. So cool to see the Taylor series expansion with sinh and cosh pointed out. I realized that if you take the derivative of any term in the expansion, you get the term to the left of it. It makes the derivative obvious. Blew me away. Thanks!
Very nice indeed! I wasn't aware of the geometric definition of the hyperbolic functions. Whilst the use of areas to define the trig functions is not quite so natural as using angles, the analogous result for the hyperbolic functions is really quite satisfying. It's worth noting that angles can't work to parameterise the hyperbolic functions as they aren't periodic, so we need a parameter than can go off to infinity without repeating points on the curve. Angles don't work for this, but areas fit the bill perfectly.
Great presentation. You may want to expand this presentation to include RF transmission line theory, and the associated hyperbolic function utilization to solve those equations.
I can't believe I finally got an explanation about what the actual fuck a sinh(x) is months after I was supposed to write an exam over it at uni by it randomly stumbling into my youtube feed
Calculating the area of A directly is relatively easy as well. Just parametrize the points in the area as r(cosh(t), sinh(t)) with r in [0, 1] and t in [0, a], then the jacobian is r(cosh(t) * sinh'(t) - sinh(t) * cosh'(t)) which happens to cancel to just r, so integrating f(r, t) = r in the rectangle [0, 1] x [0, a] we get a/2
Love it. I'm currently working on my dissertation, which heavily involves the complex exponential function, and cosh seemed to appear out of nowhere. This helps make sense of it, especially how cosh and sinh come from the real part in the same way cos and sin are in the imaginary part.
We finished Hyperbolic functions in A Level Further Math and this video is exactly what I needed. The visuals are so much help as well as the plethora of analogies with other topics, thank you so much🥹🥹🥹
As someone with A levels 55 years in my past I remember the words Cosh and Sinh and hyperbolic functions. Perhaps (no defiinitely) had I watched this video back then the dust would not have settled so thickly on my memory. So my congratulations on being young enough to have timely access to this resource.
If you want a further generalization, look at geometric algebra. It explains how you can interpret i as an oriented area, and generalize the exponential to operate on oriented planes in 3d space. This provides a nice encoding of rotations (quaternions).
Wow I'm in my first engineering year and even the professors never explained it like that Really appreciate the amount of work you've put in this video!
14:28. One little thing I want to point out is that we don’t know yet that this is necessarily true, for example, cosh(ix) could have an imaginary component which would make this comparison faulty. The statement made is in fact true and you can figure that out by representing sin and cos in terms of the exponential or by looking at the tailor series of the functions. Point is, statement is right but the reasoning given is faulty. Otherwise the video is great and gives a good intro to hyperbolic trig
Less known is that lχ| is alctually the arc length of the hyperbola from (1,0) to (cosh χ, sinh χ) when the ty-plane has the geometry of special relativity wherein, given t > y, the time elapsed along the line segment from (0, 0) to (t, y) is the square root of t^2 - y^2 (with time unit and light speed both set to 1 for simplicity). Hyperbolic angles are largely analogous in this context to circular angles in Euclidean geometry.
Note: this video starts with the analytic definition and proves that it works with the geometric one. But it's possible to go the other way around! Let's start with cosh(a) and sinh(a). We know nothing about them other than these: - The point (cosh(a), sinh(a)) is on the hyperbola x² - y² = 1. - The area traced out by this certain region is a/2. Notice that just through the geometric definition it's already possible to deduce a few identities. First: that cosh(a) is even and sinh(a) is odd. Just flip the area upside down. The x-coordinate stays put, and the y-coordinate is negated. And the other important one: cosh²(a) - sinh²(a) = 1. (cosh(a), sinh(a)) is a point on the hyperbola, so it should satisfy x² - y² = 1 by definition. The next step is to verify the integral stuff. It's the same process in the video, except we get stuck here: (1/2)cosh(a)sqrt(cosh²(a)-1) + (1/2)ln|cosh(a) + sqrt(cosh²(a)-1)| - (1/2)cosh(a)sinh(a) If there is a god then this better be equal to a/2. Here we can use an identity from earlier, just rewritten a little: cosh²(a) - 1 = sinh²(a) Then the above result simplifies and cancels into (1/2)ln|cosh(a) + sinh(a)| = a/2. A little more algebra and we get cosh(a) + sinh(a) = e^a. Here we can use our other identities: cosh is even and sinh is odd. We're forced straight into the analytic definition: cosh(a) = (1/2)(e^a + e^(-a)) sinh(a) = (1/2)(e^a - e^(-a)) Oh. And before you get suspicious about the whole cosh(ix) = cos(x) thing, plug in ix into the definition of cosh. Then cos(x) can be written as (1/2)(e^ix + e^(-ix)), and sin(x) as (1/2)(e^ix - e^(-ix)). Euler's identity makes things work out nicely in the end. Which means cos(x) = 2 has a solution, and it's i*arcosh(2). And also means that sinh(i*2pi) = 0. Not sure if it's possible to take the derivative of cosh(x) and sinh(x) without first finding the analytic formulas but considering it's possible with cos(x) and sin(x) I assume it requires some squeeze theorem
This is a flawed analysis. In actuality, it is not possible to derive analytic formulae for cos and sin from the geometric definition alone, which is why formal proofs involving cos and sin use their analytical definitions and not geometric definitions. You can derive the geometric definition from the analytical definition, but not the other way around. This is not a coincidence: if you start with the axioms of Euclidean geometry, deriving the axioms of real analysis is impossible, but you can derive all of the axioms of Euclidean geometry from the axioms of real analysis. Geometry is grounded in analysis.
@@angelmendez-rivera351 Hi Angelmendez ! 😃We meet again! Hopefully we will have a good conversation! I sincerely think there is something wrong with your conclusion. Let's think about it: The sine and cosine functions have clear geometric definitions which makes them have a certain behaviour. To each value of the angle, there will correspond a pair of values that we call cos(a) and sin(a) and those would be the coordinates of a point on a unit circle. Those values are unique! So cosine and sine are well defined functions, whatever their values at a given angle might be. If we could somehow find the relations between the values then we could have a chance to derive the analytic properties of these functions from those relations. Also, could you please give an example of a formal proof involving these functions that require their analytic expressions? (I am assuming this means their taylor series expressions??) . >> "if you start with the axioms of Euclidean geometry, deriving the axioms of real analysis is impossible," But are we really using only the Euclidean geometry axioms? Aren't we assuming something else in our geometric definitions? We are giving measures to line segments and angles using real numbers. I am not sure at all if this is allowed by the Euclidean axioms.
@@angelmendez-rivera351 This reply is about another topic. There is something about the euclidean axioms that bothers me 🧐. A point is not defined at all nor is it associated with any property, so how are we supposed to prove that the space has the topological properties we expect it to have; How do we know it is connected? How do we know it is complete? How do we know it is flat so that the pythagrean theoem holds? What does flat even mean?? How do we then define the other stuff , the straight lines and the right angles? A lot of questions , I know 😅 . Please bare with me. I hope you find them interesting as I do.
@@RU-vid_username_not_found *The sine and cosine functions have clear geometric definitions which makes them have a certain behaviour. To each value of the angle, there will correspond a pair of values that we call cos(a) and sin(a), and those would be the coordinates of a point on a unit circle. Those values are unique!* The problem is that this definition is flawed, because the parametrization of a circle is not unique at all. In fact, there exist infinitely many, and there is not a particularly intuitive list of criteria you can obtain solely from geometric concepts to uniquely pick put cos and sin out of all possible parametrizations. For example, consider f(t) = (1 - t^2)/(1 + t^2) and g(t) = (2t)/(1 + t^2). For all real t, it follows that f(t)^2 + g(t)^2 = 1. It also follows that f(0) = 1, and g(0) = 0. *So cos and sin are well-defined functions, whatever their values at a given angle might be.* They are well-defined as long as you include analytical concepts in the otherwise geometric definition. There is no purely geometric definition which makes the functions well-defined for all real numbers. *Also, could you please give an example of a formal proof involving these functions that require their analytic expressions?* If you were to prove that cos and sin are periodic functions, then you would need to either have it be part of the definition itself (which would mean you are already including an analytic concept in the otherwise geometric definition), or you would need to prove the functional-differential equation f'(x) = f(x + π/2) holds for both f = cos and f = sin, and this would require real analysis to do. *But are we really only using the Euclidean geometry axioms? Aren't we assuming something else in our geometric definitions? We are giving measures to line segments and angles using real numbers. I am not sure at all if this is allowed by the Euclidean axioms.* Well, you can have things like isomorphisms between geometric structures and subsets of real numbers, so algebra with real numbers is allowed to a limited extent. In fact, the Ancient Greeks were doing this with only compass and straightedge, limiting themselves to the so-called "constructible numbers." This was long before analytical geometry was widespread. It was, however, extremely limited, and it definitely did not have the number-theoretic power that analytical geometry has. And analytical geometry requires real analysis.
That's by far the best explanation of hyperbolic functions I have ever seen. All the others seemed ad hoc. The properties were proved, but it was never explained why the functions were considered in the first place. Everything in your video was very well motivated, thank you.
In UK further maths A level we learn Osborn's rule where the hyperbolic trig functions act the same way as normal trig functions in terms of identities etc. In the ultimate part you basically justified why this is so. It even now makes sense why you have yo flip the sign of the product of two sines as it is i^2.
why do we need to calculate the region instead of just the angle like in the unit circle where we're given the angle and able to suss out the length of the edges if i'm correct
At 2:18, i think it would have been relevant to mention that this decomposition is unique, especially for the part with taylor series. Aside of that, good video, like always
Alright. The correspondence between hyperbolic and trigonometric functions when multiplying x by i was very cool. I was not aware of that fact, but your explanation makes it seem so trivial. The problem is... I watched this video at 1am and should be going to bed. Now Im sitting here with a notebook playing around with these functions. Why do you do this to us mathematics!?
sec^2(θ)-tan^2(θ)=1 That is cooler than what you have. It leads to the cone you see in my thumbnail. Which can probably be used to do physics. I don't know if it would be better, but I'm pretty sure it can measure change in energy levels.
@@angelmendez-rivera351 have you noticed that the sum of sec(arctan(x)) and tan(arctan(x)) is quadratic? If x is a complex number, you're in C^2 territory. If you're working in 3d, that's C^3. I've seen people do RU-vid videos on C^2 with magnetism and relativity stuff.
It’s really worth mentioning that e^(ix) = cos x + i sin x is not a definition. I honestly dislike e^x notation in complex numbers because #PowersAreComplicated in complex numbers (for exponents that aren’t natural numbers). Fact of the matter is, what is meant is the application of the exponential function exp, defined as exp(x) = 1 + x + x²/2! + x³/3! + …; this definition works fine on complex inputs as well. The powers in this series are not complicated, it’s just repeated multiplication. In my Analysis I class, we have exp(ix) = cos x + i sin x by definition, because sin and cos were defined by this equation: cos x = Re(exp(ix)) and sin x = Im(exp(ix)).
Actually, that is not correct. Open up just about any calculus text, and it will state that any "proof" of Euler's identity is not accurate. Rather, it is a definition motivated by the series expansions of sinx, cosx, and e^x with x = i*theta.
I mean hyperbolic functions are called sinh/cosh has to have a reason and there has to be a connection between sin/cos and sinh/cosh. This video helped me understand it. 👍
Thanks for the interesting video! Is there any way to visually verify the equation "cosh(i x) = cos(x)" in desmos? I know desmos doesn't have complex numbers, but you can just augment desmos by adding any required operation, like multiplying complex numbers m(P,Q) = (P.x Q.x - P.y Q.y, P.x Q.y + P.y Q.x).
Intuitively, we might notice that when we compare the equations of the circle and the hyperbola, x^2 + y^2 = 1 and x^2 - y^2 = 1, that changing the y^2 from positive to negative is the same as multiplying y by i. Think of what this means if we consider a "circle" on C^2, where x^2 + y^2 = 1 for a pair of *complex* numbers x and y. We can see that the cross section across the real components of x and y of this hypercircle is a circle, and the cross section with real x and imaginary y makes a hyperbola. Thus, if our trigonometric functions extend from normal circles over R^2 to complex circles on C^2 in an intuitive fashion, we should expect this sort of identity to fall out.
@@Keldor314 Hmmm, I think it becomes rather complicated to visualize functions from C^2 to C. But I've noticed that I can verify the equation visually by plotting both cos(x+iy) and cosh(x+iy) at wolfram alpha, restricting attention to the real part of both plots and verifying that the cross section with the X=0 plane for cosh(x+iy) matches up with the cross section with the Y=0 plane for cos(x+iy), which seems to confirm the identity cos(x)=cosh(ix).
at 9:35 what does a in cosh(a) sinh(a) geometrically represent? i still dont get it. a is not an angle, but an area? examples of conversation A. in circle, i found an enemy, rotate your gun 30 degrees, then you get the enemy B. in hyperbola, i found the enemy, change your WEIRD area to 3.7/2 then you find an enemy? before you measure the required area, you will be probably already dead by the enemy or you will be 100 years old. is it possible to assess "a" as a function of vertices or foci and etc
Thank you for sharing, sir. I think it would have been more cool if the hyperbolic functions were all based on angles just like the circular functions.
13:04 wait, but didn't we define cosh x and sinh x geometrically in this segment, and are now trying to show that it's the same as the analytic definition? Then we can't bring the analytic definition's corollaries into this proof. I just wanna know
Why _YES! INDEED!_ While ultimately unnecessary, I'm going to quickly define a symbol j such that j² = 1 (1 satisfies this equation, but much like how i isn't ℝeal, j doesn't have to be either.) Much like Euler's formula, exp(jϕ) = cosh(jϕ) + sinh(jϕ) = cosh(ϕ) + jsinh(ϕ) And as a refresher: exp(iϕ) = cosh(iϕ) + sinh(iϕ) = cos(ϕ) + isin(ϕ) Now that we're refreshed on the similarities of their structures, lets now define ε such that ε² = 0 (again, it need not be ℝeal even if their is a ℝeal number that satisfies it). Here, it's easiest to use the Taylor series exp(εϕ) = 1 + εϕ + ε²ϕ²/2 + ... Since ε² = 0, every term past that is 0 leaving us with exp(εϕ) = cosh(εϕ) + sinh(εϕ) = 1 + εϕ Yup. Parabolic cosine is the constant _1,_ and parabolic sine is the identity function. Wrap your head around _that._ Then again, if they should make sense if you consider the small-angle approximation. ε can be thought of as an angle (hyperbolic or elliptic) that's infinitesimally small. The main thing that makes it questionably "parabolic" is that its "unit circle" isn't actually a parabola, but rather a pair of vertical lines as x² = 1. This can be found using the conjugate formula for the magnitude of a complex number, generalized to a hyperbolic or dual number (the official names for multiples of j and ε added to ℝeal numbers). |x + yi|² = (x + yi)(x - yi) = x² - i²y² = x² + y² |x + yj|² = (x + yj)(x - yj) = x² - j²y² = x² - y² |x + yε|² = (x + yε)(x - yε) = x² - ε²y² = x²
In the proof section, you are trying to prove that exponential definitions of sinh and cosh are equal to definitions on graph. You start with that point is equal (coshx,sinhx). You have to prove that coshx equals to (e^x + e^-x)/2 and sinhx equals to (e^x - e^-x)/2. You actually can't use these in proof because you are trying to prove them. However, in -1/2.ln|sinha + cosha| part, you used equations which you are trying to prove in order to deduce sinha + cosha equals to e^a. It can't be deduced from graph definition. This proof is invalid
What’s really interesting is that you can change the hyperbolic version of Euler’s formula into a two-dimensional analogue to the formula by using j, so that e^jx = coshx + jsinhx, where j^2 = 1 (instead of -1). It results in the split-complex plane, which has some weird geometry, like distance being equal to the square root of x^2 *minus* y^2
I wonder about their practical application, one is free hanging wire or rope (not a chain bridge), other is pursuit curve. Third is Mercator projection I think. But I never used these function in my live and they seem to be on every scientific calculator and even some advanced slide rules. All I know is they are solution to some differential equations where second derivate is the same as function. For sine it's 4th derivative and for e^x first one. But all youtube videos are about abstract concepts or identities. There must be some motivation why they exists.
The fact that you have never used these functions in your life says absolutely nothing about how many applications they have. Also, this video explains precisely why these functions exist.
Let me just ask one question ? Why don't we take the other part of the hyperbola to define the hyperbolic sine and hyperbolic cosine function ? Is the results the exact same as you obtained by considering the other part of the function?
Hyperbolic cosine comes from the x-position of the point on the unit hyperbola. Hyperbolic sine comes from the y-position of the same point. This is for a hyperbola that opens to the left and right, on the standard x-y plane.
I also made a video on the subject but with detailed computations of the integral. Check the video on my channel for more details: video title: "he Geometric Definition of the Hyperbolic Functions, and Derivation of their Formulas"
