Why do trig functions appear in Euler's formula?

Why do trig functions appear in Euler’s formula? This was the question I had when I first saw Euler’s formula. This connection between trigonometry and exponents seems so unexpected, especially along with complex numbers. To answer this question, we must journey into the intricate and beautiful mathematical relationship between trig functions, e, and complex numbers. We will look at two different ways to approach this question: one using dynamics, geometry, and the complex plane, and the other using Taylor and Maclaurin series. Both are equally fascinating, and both reach the same, amazing result by using a lot of beautiful math.
0:00 - Intro
0:38 - Unit circle on complex plane approach
7:30 - Taylor and Maclaurin series approach
12:39 - Conclusion
Additional Resources:
Geometric proof of the derivatives of sin(x) and cos(x)
ocw.mit.edu/courses/mathemati...
Animations were made by Manim, an open-source python-based animation program by 3Blue1Brown.
github.com/3b1b/manim
This video was submitted to 3Blue1Brown's SoME (Summer of Math Exposition).
3b1b.co/some1

[@sinaahadi4787]

Math and Physics are art and they are needed to perform by an artist. That was really beautiful.

[@MikeStallings2023]

First time for me that someone described it so simply and obviously.

[@longsteinpufferbatch4949]

And you not only enlightened me why e^(ix)=cosx + isinx but also why d/dx of cosx and d/dx of sinx are -sinx and cosx INTUITIVELY, so far i only had them memorized. I never knew this great visualisation before! This is gold for a high schooler like me. Please keep doing your amazing work! I like when math is this intuitive. Subbed!

[@giannisr.7733]

In our math class in uni, the teacher said we had a function A that has all the properties of the sin function, but he didn't tell us. We were talking about the sin function in it's polynomial form, and we only realised it after 3 hours of it being taught

Very well done; content- and animation-wise. My favourite video in the SoME-contest so far.

[@sardineman8994]

You put all of my thoughts about euler's Formula into a beautiful video great job

[@adleneboulebtateche156]

Absolutely stellar. I can't thank you enough for this video. Looking forward to watching new stuff.

[@godfreytomlinson2282]

This video was concise and to the point. Clear information bundled up tight.

[@jasonmaguire9425]

You've made it possible for me and I'm sure many many others to now visualise these relationships and connect the dots. Thank you so much.

[@davidm9442]

Awesome video! I've always been intrigued by the conection between trig functions and complex numbers. I really enjoyed your explanations.

[@JTtheking134]

Damn, how did you enlighten me with all this in only 13 minutes! Very underrated channel, youre so good at explaining, and you even give examples.

[@MrPabloguida]

This is some high quality material right here. I'm looking forward for your video on Fourier transform.

[@RisetotheEquation]

Great job! You made everything super clear and added some insight along the way - the best combination :) BTW - clever channel name!

[@aaqilkhan]

Amazingly done. Explanation and visualization were very well presented. Great job!

[@ravisinha312]

Looking forward to more videos! Thanks for such a lucid explanation and clear animations. Would be great if you could also share your backstory as in what goes behind the scenes to plan and create such a video. That's will make more people curious to explore manim and other tools to create more such open source videos in their domain of interest. Thanks again!

[@indocesare14]

These educational videos made with Manim are spawning everywhere lately. And I couldn't be more grateful!

[@Jaryism]

Best explanation of all vids on the internet and straight forward

[@knightace742]

This is easier than I thought it was. Thank you for explaining.

[@ClearerThanMud]

Great job -- subscribed, and looking forward to more!

[@chukwunta]

Videos like these are now my best way to learn mathematics. Thanks so much. More elbow-grease to your efforts. 👏

[@fariesz6786]

thank you! this is the first time i see a good intuitive motivation for Euler's formula _beside_ using the Taylor expansion and that always bugged me.

[@alic4631]

Beautiful man, just beautiful, I like how you started with basics

[@RuanD]

Very good! You really answered my question about that relationship and the usefulness of complex functions.

[@andresquiroga2400]

nothing new to me but still, but it completely deserves a thumbs up, these kinds of animations and explanations are always appreciated, hope you continue with these kinds of videos

[@samisiddiqi5411]

Best video on this topic I've seen.

[@longsteinpufferbatch4949]

I had the same question when i saw it recently for the first time at school. Thanks for the video :)

[@simonwax7575]

Great explanation. This I think is the essential insight of the 2 years of study I’ve just completed reduced to 15mins. Thank you.

[@johnzinhoinhoinho]

Great explanation, very clear train of thought. I wish all my teachers would be like you

[@audioplayerplaysaudio3276]

Super cool! This video of yours totally made my day/night! It's just such a good compression/combination of trig functions, complex numbers, Euler's number, and Taylor series. Of course, I have seen such contents linked in videos of 3Blue1Brown, Mathologer and others, but yours just happened to be the one tipping me over into finally GETTING IT🥳So thanks jHan!

[@alexcartwright8197]

Superb video, a work of art. Super easy to follow - you guide us well through these topics. Thank you.

[@SeeTv.]

Absolutely amazing for your first video! Question: How long did it take for you to learn Manim?

[@jHan]

It took me maybe a month to get the basics down, but it may take more or less depending on what you want to animate.

[@mihailmilev9909]

@@jHan this is your first video!? Beautiful. I'm starting to see this more and more with people who upload these pieces of amazing content for the first time as 3B1B challenge submissions.

[@indrokrisna108]

Outstanding!!!!!! clearly comprehensive

[@babujimitra7920]

Magnificent.... expect something more like this

[@GanerRL]

here before this channel blows up

[@bonbondojoe1522]

same

[@77Chester77]

Me too. I'm subscriber #89

[@Ni999]

#116 here.

[@MysteryHendrik]

#1081

[@Sarah-mp9lb]

Wow! Thank you so much for this extremely helpful video!!

[@chabanehacene]

Amazing. More video like that please

[@SayedHamidFatimi]

Love it! More content please!

[@TheJara123]

Thoughtful, beautiful and insightful...keep going because this is road not taken in the math world...and of course thanks...

[@loszhor]

Thank you for the information.

[@siener]

Great video! One of the things I don't think gets enough attention when discussing Euler's formula is this deep connection between trigonometric functions and exponential functions. It blew my mind when I realized that exponential functions are periodic on the imaginary axis and while sin and cos grow to infinity.

[@gilbertmiya4199]

Superlative. Best teachers are on KZbin!

[@skillick]

Great video, best I've seen on this topic

[@GiulioJiang]

incredibly good explanation. Every high school and university should show this video

[@mlodyG_750]

great explanation!

[@Posesso]

I liked it a LOT! Very nice channel name :)

[@dontawanpitak]

Holy smokes!!! This is amazing!!! I don't really follow the first one but for the Taylor series one, that's unreal!!!

[@janosruf2183]

Amazing animation and explanation! You have a new subscriber :)

[@culater]

Thanks for your great work 👍

[@dr.rahulgupta7573]

Excellent presentation. vow !!

[@djredrover]

really well done!!!!

[@gamalateya2413]

Amazing For years 😁 revolving around youtube to find simple explanation Finally you are 🌺🌺

[@soumenpradhan1855]

Absolutely amazing 🤗

[@jacobmacdonald223]

Best explanation I've heard yet

[@Caturiya]

Just a shortcut kzbin.info/www/bejne/a2eloZmnpbdqgdE

[@drbibhisanroy2219]

please make more content. Very high-quality sciences. Thanks a lot

[@nidalapisme]

B E A utiful! This reminds me of an 8-part video from Mr. Woo's channel explaining the same thing but he ends it to Euler's identity. Perhaps the next video from you would be explaining the most beautiful equation in the world in such a compact way. +1 from me :D

Hello, is there any email/discord to reach out to you?

[@jeremychen425]

Very good video, thank you!

[@hqs9585]

Great video!

[@thenixaless7493]

The simple way you explain this, combined with the beautiful narration is just... Even 10th grade me could understand this!!

[@TruthOfZ0]

I love this formula..its so beautiful !!

[@user-wt4lb9hg3r]

Excellent presentation. Now, discuss the derivation of Schoedinger’s equation. Your detail could clarify that. Also, you should do a segment on the natural log and complex numbers. Thanks!

[@sisyphus645]

With quality as high as this I thought you'd have over a million subscribers! Really, well done bro! Remember me when you make it big haha XD

[@annashshaha6917]

Thnks for uploading such a great video ❤💞😊

[@mighty8357]

Splendid!!

[@CuriousCyclist]

Good quality content man! A bit fast but people can pause if they need a moment to think.

[@therealowenmcbride]

Trigonometry, calculus, complex numbers, EVERYTHING is in this video😭

[@zakirreshi6737]

Well done👌👌👏.....12:08 side point yo be noted!!

[@PETERTRITSCH]

It's the first time I can figure out what the Euler equation means! And that means a lot for me!!!

[@PlanetNumeracy]

Amazing!

[@darkarz5331]

Dont know nothing about maths but i had this in recomended, guess your getting blessed by the algorithm. Looks interesting tho

[@gerardjacquemier5137]

Explication merveilleusement claire

[@DanielL143]

Excellent approach; keeping it a higher and conceptual level is the key to understanding the connections between the various mathematical concepts. Getting too lost in the details or just learning only how to calculate in a rote fashion kills understanding in favor of rigor. Both are needed. The traditional education system teaches the number crunching and kills interest in a truly beautiful language (math) by forgetting to connect all of the concepts (1) functions (2)the properties of the all important exponential function (3) derivatives (4) the application to unit vectors and the imaginary dimension that enables rotation (5) the trigonometric connection and (6) the polynomial expression of the same function using a convergent but infinite series (constraining infinity and making it work for us is truly one of the master strokes of mathematics). Then comes applications; electrical engineering and quantum mechanics which are all about waves with an imaginary component and how they sum. True understanding happens by integrating all three levels (1) the mechanics of number crunching which allows us to speak the language (2) the high level conceptual connections between various mathematical topics and approaches which validates the consistency of the language and (3) the application of mathematics as a tool for modeling systems, solving problems, optimizing and evolving systems and Finally there is the mystery that surrounds the fit between the model and the system and the misfit between GR and QM and something deeply hidden. Beauty and mystery, it doesn't get any better. Thanks!

[@BRYDN_NATHAN]

Thank you. Favorite number 👍

[@Ni999]

_Excellent._

[@2001ayham]

Hey great video! I'm studying Electrical engineering and this was very interesting for my signals course.

[@carultch]

I like how Heaviside's Pi and Lambda function are named by symbols that look like the shape of the signal.

[@leeris19]

HELLA BEAUTIFUL!

[@rajanvk939]

Wow excellent explanation. Could you please 🙏 make videos on Vector Geometry

[@kanhaiyalalrajput3215]

Great Job 👍👌. I needed this explanation.

[@MarcoMate87]

Very well done video, and excellent explanation. There's another proof for the coincidence of f(x) = e^(ix) and g(x) = cos(x) + i sin(x) for every real x. These two functions both solve the Cauchy problem y' = iy with y(0) = 1. As the solution of this problem is unique, f and g must be equal everywhere.

[@codatheseus5060]

I love using (x+y) instead of just x in my Taylor series. You gotta double the number next to the factorial to keep it good

[@darthvitiate7373]

ngl thought this a was a 3blue 1 brown video then i saw the channel name keep up the good work

[@pokemonxyzseriesunofficial206]

U know what. You should make more of it.

[@Spongman]

very good. although, some of the manim latex transitions could be redone to minimize the amount of text that changes. eg @12:24 only the 'cos x' part needs to change, but the whole equation goes through the mangling transition which hides the fact that it's only the real part on the rhs that's changing.

[@SD-mc9xm]

❤️ you explain it so beautifully, lol you remind me of 3b1b

[@adlg5158]

This video made Euler's identity the clearest to me, how do you not have more than 50 subscribers?

[@BRYDN_NATHAN]

Thank you. The dangle has an angle. 👍

[@ILSCDF]

Beautiful

[@manfredbogner9799]

very good

[@dankdungeon5104]

Insane video

[@miro.s]

Very clear speaking and graphics. Only derivatives of e^(ix) are discrete so at that point your proof is wrong. You should show that derivatives can be uniquely extended to fractional derivatives. Then that the extension is smooth. Then that fractional derivatives of e^(ix) don't change absolute value of the function. Then you finish the proof showing that e^(ix) = cos x +i sin x

[@angelmendez-rivera351]

The statement that derivatives can be uniquely extended to fractional derivatives is incorrect. In fact, it can be rigorously proven that over any vector space, a linear operator which is surjective and has nontrivial kernel cannot be fractionalized, even if you disregard the ambiguities that inherently emerge with fractional exponentiation in the scalar field.

[@CoupleMadness]

You are amazing sir 🥰 , I love Mathematics ❤️ so please do a favour for me keep making such amazing videos ❣️ Love from India 😍

[@thomasolson7447]

I've been playing with the Lorentz Factor. e^(i*arctan(i*v/c))=(-v/c+1)/sqrt(-v^2/c^2+1) which is γ*(1-v/c).

[@H3XED_OwO]

11:53 -12:07 it's all coming together! 🤯

[@hichamhenna8011]

At time 06:25, he tells us that: (the derivative ie^(ix) has no real constant changing the function ==> this means that the magnitude of the derivative stays constant at 1); this statement that I wrote between brackets it is not as intuitive as I wish. Further explanation please!

[@VictorMasnikov]

Hey, love this lesson! Now i can create more complex fractals than ever, thanks!!!

[@kiepier]

At 9:33 does cosθ = dy/dθ because the triangle with θ at the origin is similar to the triangle with θ on the unit circle? I guess it makes sense if the magnitude of the rate of change is constant like e^ix.

[@621Tomcat]

I looked at the thumbnail and thought it was a 3b1b video Edit: read the description, now I know why Also edit: this video was very beautifully made

[@curtpiazza1688]

Great STUFF! 😂

[@rajinfootonchuriquen]

A think the first explanation needs at least to understand curves in space and their derivatives (vector fields), but the second only needs basic differential calculus, so the second is a better approach i think for explain it. I like the fact, using linear algebra, that the exponential function is the eigenvector of the differential operator for or eigenvalue, and then a second-degree differential operator has as eigenvector the trig. functions with eigenvalue = -1, so the trig. function must be a linear combination of exps; then the fact that the linear operator is degree two, so the eigenvalue of that operator corresponde to the square of the eigenvalue of the first-degree operator, tales that the eigenvalue of the linear eq. D^2(y) = -y it's just "i", and then your initial conditions dictate the linear combination of exponential functions. That result requieres to know linear algebra and calculus, but for me it's the less "magical" because you are not matching what it seems pears and apples, or just pluging "i" in exp because someone was curious.

[@fifaham]

Truly amazing discovery - what is even more amazing is the human brain who invented the magical J = SQRT(-1) and found its correlation to trigonometry. It is all hidden inside our brains, and manufactured by the Universe !