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:
https://www.jstor.org/stable/2974881?...
https://arxiv.org/pdf/1610.04825.pdf (equations of the involutes)
https://math.stackexchange.com/questi... (the StackExchange post)
https://en.wikipedia.org/wiki/Involute (More about involutes)
https://support.google.com/youtube/an... (Advertiser-friendly guidelines)
https://www.unicef.org.uk/donate/dona... (UNICEF Ukraine Appeal)

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

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https://forms.gle/QJ29hocF9uQAyZyH6

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

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