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)

Animations were made by Manim, an open-source python-based animation program by 3Blue1Brown.

This video was submitted to 3Blue1Brown's SoME (Summer of Math Exposition).

[sinaahadi]

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

[chukwunta]

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

[gilbertmiya]

Superlative. Best teachers are on YouTube!

[MikeStallings]

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

[giannisr.]

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

[longsteinpufferbatch]

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!

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

[alexcartwright]

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

[godfreytomlinson]

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

[sardineman]

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

[Jaryism]

Best explanation of all vids on the internet and straight forward

[indocesare]

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

[JTtheking]

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.

[drbibhisanroy]

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

[GiulioJiang]

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

[audioplayerplaysaudio]

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!

[alic]

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

[MrPabloguida]

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

[babujimitra]

Magnificent.... expect something more like this

[andresquiroga]

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