The geometric interpretation of sin x = x - x³/3! + x⁵/5! -...

We first learnt sin x as a geometric object, so can we make geometric sense of the Taylor series of the sine function? For a long time, I thought it was just my dream, but actually, it is not!

This proof only uses very elementary methods, and depending on your definition of calculus, doesn't actually use calculus. The most we are using is a limiting process, and definitely no differentiation or integration here.

This proof is very beautiful - not only that it unveils the geometric meaning of each term in the series very beautifully, but also understandable by a normal high-school student with a little bit of patience. I am very surprised that it has not appeared on YouTube before, and even if it does exist on the internet, it is far too unpopular, and so I have to bring this up!

Obviously this is not my proof. See the sources below.

Sources:

Music:
Asher Fulero - Beseeched
Aakash Gandhi - White River, Heavenly, Lifting Dreams, Kiss the Sky

Video Chapters:
00:00 Introduction
00:50 Preliminaries
02:10 Main sketch
06:03 Details - Laying the ground work
09:42 The iteration process
11:11 Finding lengths of involutes
14:57 What? Combinatorics?
18:44 Final calculation
20:45 Fundraiser appeal

Other than commenting on the video, you are very welcome to fill in a Google form linked below, which helps me make better videos by catering for your math levels:

If you want to know more interesting Mathematics, stay tuned for the next video!

SUBSCRIBE and see you in the next video!

If you are wondering how I made all these videos, even though it is stylistically similar to 3Blue1Brown, I don't use his animation engine Manim, but I will probably reveal how I did it in a potential subscriber milestone, so do subscribe!

Social media:

For my contact email, check my About page on a PC.

See you next time!

[mathemaniac]

I might change the title to "Without calculus..." because... how else do you combine "differentiation and integration" so that the title is snappier?

The only thing that held me back from making this video a fundraiser is the fear of political comments, or any comment that seeks to create conflict, so please do me a favour by not making those comments. Anyway, donate if you can; if not, like, comment and share this video so that this gets more people’s attention.

This video is a lot slower than my usual videos, but it might be better for understanding. If somehow you think this is too slow, you can always speed it up. The hope is that you don’t need any university-level maths to understand this video.

NOTE: In the last part, k is supposed to be fixed, so each bracket do go to 1 when n tends to infinity.

[anwyl]

This feels like tricking kids into eating vegetables. "Oh, taylor series is too complicated? Okay we'll just add up a bunch of lines tangent to curves!" I love it.

[henrymarkson]

Your content rivals that of 3b1b and there is no greater praise in the world of YouTube maths.

[nathanwestfall]

This is what I needed 20 years ago! A very satisfying approach that gives meaning to an otherwise non-intuitive result. Keep it up! :-)

[blackpenredpen]

2:59 😮
thanks for an amazing video!!!

[glarynth]

Great illustration! Also, e^x is hidden in there, as the total length of the spiral (including the horizontal segment from the origin to (1, 0)). This shows an intimate connection between the exponential and the sine/cosine, which typically isn't apparent without invoking the complex numbers.

[timermens]

Very nice. Sice 3Blue1Brown and Mathologer their "moving average probability" of uploading content seems to be in a downwards trend, this might be the next channel showing the beauty of math. Pls keep doing these kinds of videos!

[yashrawat]

Woah the proof and its details are too good ( the entry of Pascal's Triangle and Binomial Coeffiecients too )

[physicsboy]

This is an interesting video and the showing of the proof is simple enough that even I can understand you explained it well

[nomanbinmorshed]

Another beauty of these videos are that: They make u feel jealous in a very positive way and u start thinking as to why I couldn't think of this masterpiece!!! Love for yet another beautiful channel

[yakovify]

While the Pascal part is quite beautiful I think it abstracts away a little too much from the geometrical intuition. We can simply note that the 'rate' of unwrapping of involute n+1 is proportional to the distance from the point along the involute n. Thus suggests that the distance of involute n+1 at the unwrap point is the integral of the previous involutes distance from the point of interest.

Thus a recursive arguments makes the nth involutes have a distance function x^n/n!

[coffeecup]

More of a "How to sneak calculus into a 'without calculus' explanation" video

I love it, keep it up. There are a lot of people who will have a better intuition for calculus when they get to their intro class if they watch this stuff.

[kbin]

This was incredible. I came here with no expectations of understanding more than half of the proof (happens to me very often with these kind of videos), since I have no superior education yet, but I was amazed by how clear everything was, only in the last part I had to pause it for a while to understand

Thanks for making your videos so accesible man, greetings from Perú

[rms_txrx]

This is precious, a gem of mathematical insight. You explained everything very carefully, so much that
by 1/3 or so of the video I realized where you were going - and yet I chose to keep watching, because… DAMN! You’re good!

Keep it up and you might be making videos like 3b1b in no time! I’ll wait for that!

[kshitijthakkar]

This just absolutely blew my mind mate, Keep up this great work. This video is just an exquisite example of how beautiful, connected and simple pure math can be. Congratulations for pulling this one. All the best for your future ventures ❤️❤️❤️👍.

[titan]

This is the very first video of yours that I've watched, and I've already subscribed to your channel because this one is, by far, the best explanation I've ever come across with why sinx is expressed the way it is. You are an excellent teacher. Thank you!

[codewordbw]

When you see videos like these and you understand mathematics to a certain degree, it's always awesome whenever you see a proof that is going in a direction that you've never seen before. But the best part is when you're partially through the proof and you can use your past knowledge to see where the proof is going and it finally clicks as to why the proof works. It's so cool, every time

[anhthiensaigon]

dude this is so perfectly well done that it seems like a weapon by itself already.

[MichaelRothwell]

Congratulations on bringing a geometric perspective to something so unexpected as a couple of Maclaurin series - and throwing in a favourite result of mine about the sums of columns of Pascal's triangles as a bonus! The whole approach was very clear, so I could see fairly early how this was going. The whole thing was a pleasure to watch, which is quite an achievement considering how much was going on here mathematically.

[ab]

Your videos are always a pleasure to listen to, even the slower ones! Well done for bringing a geometric POV to an algebraic phenomenon.