Why hyperbolic functions are actually really nice

Today we unpack everything to do with hyperbolic functions. In calculus we often see an analytic definition of hyperbolic cosine and hyperbolic sin expressed as exponential functions. But why? Analytically, this is just asking for the even and odd components of e^x. However, we can connect this geometrically to hyperbolas, and specifically to the area enclosed by a hyperbola.

0:00 Even and Odd Functions
2:24 Analytic Definition of cosh and sinh
3:19 Graphic cosh and sinh
3:53 Taylor series and derivatives
5:12 Hyperbolas
6:42 Defining trig functions geometrically
9:22 Defining hyperbolic trig functions geometrically
10:45 The geometric and analytic definitions are the same
14:03 Euler's Equation

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[kasiphia]

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

[qubex]

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.

[nicholascooper]

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!

[jamesmnguyen]

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.

[chrisgreen_]

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

[Jurgan]

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.

[Dr.Cassio_Esteves]

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.

[Ninja]

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.

[kimjong-du]

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!

[andrewbekhiet]

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!

[lordforlorn]

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

[AdamHoppe-bmlr]

I love this man, I've almost completely abandoned my suggested lectures for your videos.

[strangeWaters]

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).

[wargreymon]

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

[birjeetbrahma]

The way you speak every topic is really heartwarming.😊

[alphamf]

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

[arbodox]

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.

[TimVT]

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!

[Carpirinha]

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

[Kralasaurusx]

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.