In school we are rarely ever taught the connection between the hyperbola and sinh(x), etc...Very interesting.

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.

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.

It's these kinds of videos that make mathematics actually interesting.

This really is a superb introduction to hyperbolic functions. All of the key ideas in 15 minutes explained perfectly!

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!

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.

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!

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!

This is crazy! We were never taught that in school. It makes so much sense

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).

The way you speak every topic is really heartwarming.😊

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!

Like that approach starting with odd and even function, easily one of the best video on hyperbolic functions

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

Very nice presentation, thank you.

That was amazing connection. Thank you

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.

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.

Great video! I want more videos continuing explaining this now!

Hi Dr. Bazett! So cool!

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 like them because they're like circular trig functions, but stretchy!

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.

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

U could also have talked about the explicit formula for the inverses of Cosh and Sinh !

Most beautiful math explanation

lovely video!

excellent video❤

great vid!

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.

Thanks! Good explanation …

Sir please come up with a series on Numerical methods for ODE & PDE.

Amazing and the best introduction to Hyperbolic I was thinking about hyperbolic functions a day before or so and this video came out! Coincidence ?

Love you Prof!⭐

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.

Superb !! 👍

Also sir, considering that youre a maths professor.. Could you please make a video on statistics for machine learning

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🥹🥹🥹

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. 👍

You are a great math sorcerer.

Excellent.

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

Enjoyed the math, but also - what a great T-shirt!

12:17 variable “t” is introduced out of nowhere and gets substituted for as though it was x later, is that a mistake, should that be x instead?

I wish I could give 2 thumbs up to that great video. Such great content!

Hello sir, been subscriber to your channel since sometime..love the content..thanks for uploading.. Lots of love and appreciation from india 🇮🇳

Subscribed 👍

Thanks!

Make a playlist about complux analysis please😢

Thanks

where was this when I was in my first year of astrophysics TvT still I really enjoyed this video

Cool, like this video already and i haven' t even see all of it:)

Great!

What is the aplication to create the video?

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

That's the most beautiful thing about mathematics, isn't it?

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

For special relativity this is a God sent

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).

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

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)).

what kinds of tools do you use to generate such plots/videos?

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!?

If we habitually moved close to light speed, this would be so intuitive

At 6:15, x=tan(theta) and y = sec(theta) gives x²-y²=-1. You said it the other way around.

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?

Professor , in 6:14 , x = sec(theeta) and y = tan(theeta) isn't it ?

Your new Subscriber

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

Dear Dr could you please proof lambert w function formula w(xe^x)=x

Hi, I do not get 14:30 argument. like how 2+3 == 4+1 can result to 2=4 and 3=1 !!!

12:32 t should be cosh(a). From-to notation was a little bit messy. x = 1 to cosh(a) applies to both terms.

Great

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.

Is there also something like parabolic trig functions?

started sounding like Fourier series

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.

Something feels so circular about this and many other video that does similar things with Hyperbolic trig. Is it circular logic?

6:12 Tried plugging it in and it's -1. So you had it backwards. x should be sec theta and y should be tan theta.

I NEVER learned this in school. I had to research this independently... I don't know why they omit the derivations smh

And the "Hyper-Fourier" transform for all this. Maybe another video?

I love your shirt.

This is where I think we need slow down , really slow down just like Fourier and Laplace transformer

12:30 i think there is a t instead of a x, else i dont understand :D

I just changed my major from Finance to Maths!!

Holy moly that was some sexy af math

Remember folks, radians, not degrees.

Can we call y=x a parabolic sine, and y=x² a parabolic cosine? 🤔

Muito bom. Tô pensando em fazer meu TCC em funções trigonométricas de números complexos e suas relações com funções trigonométricas hiperbólicas.

So that’s where that function from this week’s tutorial comes from 😅

Leibniz's calculus > Newton's calculus

Thanks for your video! You slow down your speech. You are running now! Just speak in a normal speed as a lecturer would do in his classroom.

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"

Could you teach me where you bought that fucking beautiful T-shirt?

🤠

Why do you only deal with the real numbers? This gets a lot more general when one proceeds to complex numbers, and above.

The complex part seemed like the biggest deal