Law Of Cosines I (visual proof)

This is a short, animated visual proof of the Law of Cosines using Thales triangle theorem and the intersecting chord theorem. This theorem relates the side of a triangle with the other two sides and the angle between those two sides. #mathshorts #mathvideo #math #trigonometry #lawofcosines #triangle #manim #animation #theorem #pww #proofwithoutwords #visualproof #proof #circle #thalestheorem #intersecting chords
This animation is based on a proof due to Sidney H. Kung from the December 1990 issue of Mathematics Magazine (doi.org/10.230...) page 342.
To learn more about animating with manim, check out:
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Background music:
The Temperature of the Air on the Bow of the Kaleetan by Chris Zabriskie is licensed under a Creative Commons Attribution 4.0 license. creativecommon...
Source: chriszabriskie....
Artist: chriszabriskie....

[@boredomgotmehere]

Another wonderful and brilliantly visualized mathematical theorem. You’re one of a kind&mind! Thank you.

[@MathVisualProofs]

Thanks! This is an old one. Some things I’d do differently now :)

[@thiagosdev]

For anyone wondering about adjacent is 2acos(θ): cos(θ) = adjacentSide / Hypotenuse hypotenuse * cos(θ) = adjacentSide hypotenuse is 2a, so: 2acos(θ) = adjacentSide

[@meD-q3k]

thank you just save me from 2 hour of confusion

[@ariffdanial8423]

I make my proof to understand this. Why not i read comments first😂. I waste 1 hour haha

[@xxii9911]

thank you so much

[@jakobthomsen1595]

Great to see an application of the Thales / Inscribed Angle and Intersecting Chords theorems!

[@MathVisualProofs]

I've got a playlist about Thales featuring all the videos that uses that theorem (or inscribed angle as you note). I also have a proof of intersecting chords fyi: kzbin.info/www/bejne/n2Ktd6tjZdJ7j68

[@jakobthomsen1595]

@@MathVisualProofs Thanks, will take a look!

[@math2693]

This is brilliantly amazing 😍😍😍

[@MathVisualProofs]

It’s definitely one of my fave proofs of this fact.

[@ll-lw8vn]

Genius, my precal didn’t even explain why, she just show us the equation and let us remember it. This is so useful

[@mohammedmarhnine8194]

why didn't we see this kind of channel on top, gosh, you are amazing, thank u

[@MathVisualProofs]

Thank you for watching!

[@PriyanTekchandani-t6i]

Thank you for this, i have been trying to understand trigonometry, and this simplified things a lot

[@MathVisualProofs]

Glad it helps!

[@PriyanTekchandani-t6i]

@MathVisualProofs can prove law of sines visually too btw? That would help too I mean.

[@MathVisualProofs]

@ I have one on my channel. I have other trig ones too (other law of cosines for instance). Check my trigonometry playlist

[@PriyanTekchandani-t6i]

@MathVisualProofs do you do linear algebra too?

[@MohammedRayaanPasha]

Elegant ❤

[@MathVisualProofs]

😀👍

[@sumedh-girish]

Simply brilliant, it is these visualisations that make the math in my head usable. Thank you !!

[@MathVisualProofs]

Thanks for checking them out!

[@pardeepgarg2640]

1st And thx for such a wonderful animation

[@MathVisualProofs]

Thanks for continuing to check them out :)

[@WilsonianGarveyite]

Came for the math, stayed for the music.

[@abbasfadhil1715]

So poetic

[@minimalrho]

Is there a modern book that covers all the proofs of Euclidean geometry from say Hilbert axioms that has visuals as well (so excluding Hilbert)? It occurs to me that I've never seen a proof of Thale's theorem and the like.

[@MathVisualProofs]

This is a good question. I don’t know the answer. I’ll admit that my knowledge of geometry from the foundations is not great. For instance I only relatively recently learned that the Pythagorean theorem is equivalent to the parallel postulate. This means that Thales theorem and PT are equivalent too. One day I’ll prove Thales using PT on here so that I have visual proof of the equivalence (already have Thales -> PT and actually it’s hidden in this video too :) )

[@Scigatt]

Probably the best option for you is Geometry: Euclid and Beyond by Hartshorne.

[@PriyanTekchandani-t6i]

The Elements, written by Euclid himself.

[@vincealbertomateo4636]

waaaao! thank you

[@nikitas3729]

Amazing

[@MathVisualProofs]

Thanks! This is one of my fave proofs of this fact.

[@nikitas3729]

@@MathVisualProofs There is also a good proof with drawing squares on the sides and using areas with heights

[@MathVisualProofs]

@@nikitas3729 Definitely! I have it in the queue at some point :)

[@gamePlays9699]

Very NICE!!!!!!!!

[@MathVisualProofs]

Thanks!!

[@neytirin9747]

thank you sir

[@MathVisualProofs]

Glad it helped. Thanks for check it out!

[@TheBollibolli]

Finally found this!

Very nice animation. What if theta is bigger then 90 degrees?

[@MathVisualProofs]

Thanks! Yes. Unfortunately this diagram only works for acute angles. I have other law of cosines videos in the queue that should work for other angles. And you can check out this one from @jacobroggy kzbin.info/www/bejne/poSbgpign6l4Zsk

[@kinggaming9437]

This will definitely help me in real life and pay my bills

[@harley6659]

I don’t get why the leg adjacent to θ is 2a cos(θ)

[@MathVisualProofs]

The cosine of an angle is the ratio of adjacent leg over hypotenuse. So the adjacent leg length is the hypotenuse times the cosine of the angle.

[@ThefamousMrcroissant]

sin(a) = opposite/hypotenuse. I.e. hypotenuse=sin(a)*opposite. The opposite side is 2a.

[@bikashchakraborty8816]

Nice.👌

[@MathVisualProofs]

Thanks :)

[@hanimahdi7244]

nice!

[@MathVisualProofs]

Thanks!

[@Youtube..Enjoyer]

Ok fine you convinced me

[@eipimath]

Thanks for sharing! I posted a short video on deriving the law of cosines, It should apply for all angles (acute, obtuse, reflex, negative). Hope to get your thoughts.

[@MathVisualProofs]

Nice one! I enjoyed watching yours. Also you might want to check this one out : kzbin.info/www/bejne/poSbgpign6l4Zsk from @jacobroggy

[@carlosdelossantos5115]

How is the animation done?

[@MathVisualProofs]

This is done using ManimGL.

[@Deepindreams1]

How is leg adjacent to the angle tetha equals to 2a.costetha

[@MathVisualProofs]

The cosine of an angle is equal to the length of adjacent edge divided by length of hypotenuse. So multiply both sides by hypotenuse, which is 2a to get adjacent edge by itself.

[@justunlocktheuniverse]

Just being curious, how do you make animation like this?

[@MathVisualProofs]

These are made with 3b1b's python library called manim.

[@justunlocktheuniverse]

@@MathVisualProofs thank you!!

[@justunlocktheuniverse]

@@MathVisualProofs your videos are absolutely amazing ,it helps me alot to understand concepts easily and beautifully :)

[@MathVisualProofs]

@@justunlocktheuniverse Thanks for watching! I am glad they help. :)

[@Withmusclecutie]

But what if c was bigger than a and b?

[@GourangaPL]

Al-Kashi's law of cosins :)

[@PedroAssumpcao-on6db]

This must have come from The Book

[@blitzvii322]

what on earth is going on

[@edilsonmachado1961]

legal

[@khanohere-lp9fp]

BHEEN KA LOWDY AGER TU TRIANGLE SHI BANATA TO MUJJA 2 GHANTA NA LAGTY SAMJHNY MA BAKI THANKU OSM VEDIO

[@ckshene7212]

Sorry. This proof was due to Sidney Kung, which is incomplete. What is angle \theta is an obtuse angle or the circle you draw does not contain the triangle. While the general idea still works, but the reasoning would be a little of different. Please stop spread this incomplete proof. Additionally, many Proofs without Words share the same deficiency.