Sine of a Sum II (visual proof, trigonometry)

This is a short, animated visual proof of the sum formula for the sine function using the Side-Angle-Side parallelogram area formula. This theorem relates the sine of a sum to the sum of products of sines and cosines. At the end, we provide a hint for how to use the same proof with different labels to prove the cosine difference formula. #mathshorts #mathvideo #math #trigonometry #sine #sinofsum #triangle #manim #animation #theorem #pww #proofwithoutwords #visualproof #proof #circle #sideangleside #parallogramarea

Thanks to Phil Boswell for noting a mistake in my coding that led to off-dimensions!

Here is an alternate proof of the same fact:

And here is a visual proof justifying the area formula for a parallelogram:

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This animation is based on a visual proof by Volker Priebe and Edgar A. Ramos from the compendium Proofs Without Words II (More exercises in visual thinking) from Roger B. Nelsen (page 40.)

To learn more about animating with manim, check out:
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Music in this video:
Creative Commons Attribution 3.0 Unported License

[MathVisualProofs]

Thanks to Phil Boswell for noting a mistake in the coding! This version should be a bit more accurate :)

[NebelmondDso]

in my opinion the most intuitive and easy to understand proof, great animation, well done!

[Mathematica]

That was an epic demonstration! Beautiful.

[bachirblackers]

Very beautiful visualisation . Thanks .❤

[kdicus]

That's exactly what I was looking for. Absolutely beautiful. You mathematicians using this new canvas is the same as what Euclid did with paper. That was amazing. This is better. Gorgeous work. Thank you.

[tsuwako]

Great proof, I like the epic music there !

[oosmanbeekawoo]

Hello. Kind advice. Choose a nicer set of colours for the geometries. The background maybe. I advise a harmonic colour palette that you can find in graphic design or art manuals, like pastel tones. Or create your own, following rules of harmony in colours, like Triadic, Monochrome, Tetradic, things like that. They really contribute to the aesthetics.

[updown]

Interesting 😀 There are two problems (not with a video, with me haha): I am not English/American and I didn’t got this in my math class yet 😅 But still it’s very nice to watch and may become helpful in future 🙂
Greetings from Poland mate! 💪

[vlogabe]

But why the sin(a+b) is the unshaded area? 00:43

[pa]

This fucking rocks. Now vary alpha and beta!

[Theo-kilu]

dude your music is too dramatic, i think it’s more fitting something classy, like the videos from “think twice”