Tangents of Pi/8 and 3Pi/8 (visual proof without words)

This is a short, animated visual proof demonstrating how to compute the tangents of Pi/8 (22.5 degrees) and 3Pi/8 (67.5 degrees).
If you like this video, consider subscribing to the channel or consider buying me a coffee: www.buymeacoffee.com/VisualPr.... Thanks!
Check out these related videos about trigonometry:
• Tangent 75 and tangent... (tangents of 15 and 75 degrees)
• Half Angle Tangent For... (half angle tangent formula that you can use to get this result)
• A Five Pi Diagram (pi ... (arctangent reulsts)
This animation is based on a visual proof attributed to Grégoire Nicollier from Sion, Switzerland by Alexander Bogomolny (see www.cut-the-knot.org/pythagor... for details and relation to folding A4 paper).
To learn more about animating with manim, check out:
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#mathshorts​ #mathvideo​ #math​ #trigonometry #mtbos​ #manim​ #animation​ #theorem​ #pww​ #proofwithoutwords​ #visualproof​ #proof​ #iteachmath #squares #pi #triangle #rectangle #root2 #squareroot #22.5 #67.5 #isoscelestriangle #tangentfunction #tangent
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Music:
Glass Wall by ADERIN | / andrei-burcea-20972653
Music promoted by www.free-stock-music.com
Creative Commons / Attribution-ShareAlike 3.0 Unported (CC BY-SA 3.0)
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[@Tosi31415]

you could further simplify the second one into sqrt2+1

[@MathVisualProofs]

👍

[@Ninja20704]

We can also get sin and cos from this diagram, tho they are not as neat as they involve nested square roots.

[@LittlePaimon]

Thanks to the video's visual evidence, my students were deeply impressed！

[@tcaDNAp]

I'm here after hearing the latest episode of My Favorite Theorem, and I'm impressed by the huge number of animations!

[@tcaDNAp]

Also, totally unrelated, but The Map of Tiny Perfect Things is a cute romcom that also pairs Groundhog Day and visual proofs

[@MathVisualProofs]

Thanks for coming to check them out!

[@MathVisualProofs]

I’ll check out the movie :)

[@Math-Lightning]

Your all videos are perfect. I liked all of them.

[@MathVisualProofs]

Thanks!

[@yplayergames7934]

This, is, really, beautiful

[@MathVisualProofs]

Thanks!

[@Inspirator_AG112]

I also have also used an octagon before to prove tan(67.5°), and my classmate from last school year extended that to tan(75°) using a dodecagon.
I have a post with around 75 upvoted on r/mathematics showing this (I am u/InspiratorAG112). I will try to put the post ID in the replies (since CZcams disallows URLs in comments by default).

[@Inspirator_AG112]

*Post ID:* 12okz9r

[@ikiyekatlananparsomen]

your videos are pretty awesome. seeing that all these proofs visually makes mathematics more enjoyable and sensible. however I must add that you should choose the colors better, I didn't manage to see the isosceles triangle at my first watch. also that would be nice if you could add voice narration to explain what is going on, that will make videos easier for audience

[@MathVisualProofs]

Thanks. I worked hard to find a color schemed that should be accessible (davidmathlogic.com/colorblind/#%23332288-%23117733-%2344AA99-%2388CCEE-%23DDCC77-%23CC6677-%23AA4499-%23882255). I get that the blue in this one isn't quite as good of a contrast on the white. I also appreciate that you would prefer narration. I do wordless versions less than half the time, but for certain videos, like this one, I am not sure what narration can add... as the goal might be to keep people thinking about why each step works and how they might get similar results :) Also, hard to find narration time.

[@ikiyekatlananparsomen]

@@MathVisualProofs this idea is good too :), thank you for replying 👍

[@DanGRV]

Also, 1/(sqrt(2)-1) = sqrt(2)+1

[@Hexhexhexhexhexhex]

can you please solve this sum problem i found
1^n + 2^(n-1) + 3^(n-2) + ... + (n-2)^3 + (n-1)^2 + n^1 = ?

[@phiarchitect]

Pi's irrationality is irrational.

[@Navigator87110]

Would rather hear narration than that... "music." Just sayin.

[@MathVisualProofs]

Understand. Unfortunately some times I just can’t find the time and with diagrams like this I’m not sure what speaking would add. The fun of this one to me is thinking through wholly all the angles and sides are what they are. But I take your point. Thanks!

[@Navigator87110]

@@MathVisualProofs eh, sorry, didn't mean to be a jerk about it.

[@MathVisualProofs]

@@Navigator87110 no not at all! I appreciate the feedback and knowing what people prefer. I was just trying to justify my choice here :)

[@Geek37664]

Now do it with an octagon…

[@MathVisualProofs]

Which one: mydigitalpublication.com/publication/?m=53548&i=787665&p=24&ver=html5

[@Geek37664]

@@MathVisualProofs I’m partial to the second method, but they all look equally cool.

[@Inspirator_AG112]

I actually have one, somewhat different from what the uploader linked (using a sum of sines), on an r/mathematics post titled: "Tangents of 67.5° and 75° obtained with polygons."
(The post ID is 12okz9r. Also, if you see u/InspiratorAG112, that is me.)

[@MathVisualProofs]

@@Inspirator_AG112 very cool!