Law of Cosines Visual Proof: the “Loctagon”

In this video, I derive my visual proof for the Law of Cosines. Similar to the popular Pythagorean Theorem proof, this proof also compares the areas of two figures. The goal of this video is to provide an intuitive explanation for the existence of the trigonometric term in the Law of Cosines. I assume, if you're watching this, that you are far enough in your math education to know what the Law of Cosines is and how to use it. That is, you should know high school Geometry and some basic Trigonometry.
I came up with this generalization myself a few months ago. As I was making this video, I saw another "area comparison" proof on Wikipedia. However, I think the one in this video is different enough in its construction (and, frankly, more beautiful) that I can still claim that it is original. Please let me know if it isn't!
I want to thank Grant of 3B1B and everyone involved in the development and documentation of Manim CE, which I used to make this video, as well as everyone organizing the Summer of Math Exposition. I also want to shout out Micro Math Visual Proofs, whose videos inspired me to go looking for more visual proofs.
This is my first time making anything with Manim, and it probably shows. Regardless, I hope you all enjoy watching!

[@levprotter1231]

Fantastic! There’s something oddly 3D feeling about those animated shapes..

[@matthewbridgeman57]

Honestly this feels almost *4*-dimensional. Can’t put my finger on why, but it’s just so smooth that it looks both natural and unnatural at the same time!

[@1ab1]

Truly impressive! Hard to believe this gem was hidden for >2000 years.

[@johnchessant3012]

That's brilliant!! Definitely cooler than the standard proofs (coordinates or drawing an altitude) and I really like how you discovered it by building on the already beautiful proof of the Pythagorean theorem

[@jakobr_]

I may be *very* biased but I agree. The law of cosines needed an intuitive proof where its “length x length” numbers actually correspond to areas. I’m convinced that the law of cosines is naturally a two-dimensional statement. This is one of the only few proofs that acknowledge this nature.

[@jessereed1736]

A great video to visually see the proof of the law of cosines, it is one of my favorite proofs as well. My professor going over it was what helped me realize I might want to major in math.

[@eterty8335]

the animations on this video feel like someone has turned 4D into a visually understandable concept. people who studied geometry and came up with this stuff back then must have been literal gods...

[@MathVisualProofs]

Excellent work here! Very cool proof. I enjoyed it. And thanks for the shout out :)

[@jakobr_]

Thank you, I’m really glad you liked it! I’d been watching your videos for a while before deciding to investigate this equation. None of the proofs I found were quite to my satisfaction, so I went and found my own, motivated by the idea that there *must* be a beautiful way to show this truth, just like in many of your videos. The shout out is deserved!

[@MathVisualProofs]

@@jakobr_ Hey, do you have a contact email? Or can you email me?

[@jakobr_]

@@MathVisualProofs I just sent a message to what I *think* is the right address. If you didn’t get anything let me know

[@tucan1309]

this is amazing, I love these proofs so much where visuals are used instead of endless equations, of course rigor is needed as well but love seeing these proofs

[@jakobr_]

Without rigor there’s always the slightest possibility that I might be tricking you with the visuals. The next best alternative to rigor (and a more entertaining one in my opinion) is to give a plethora of data points in the form of that ending animation. I’m glad you enjoyed the video!

[@rajinfootonchuriquen]

Visual proof are fallacies, but they are pretty

[@Roonasaur]

@@rajinfootonchuriquen Whoa dude . . . "unexacting" is a far cry from "fallacy." Lighten up, geez

[@puppergump4117]

@@rajinfootonchuriquen You just destroyed graphs, what are we gonna do now

[@narfharder]

Hi stranger! You should know this gem's going in my special "edu" playlist, to make sure I can find it again. 1:41 "That's not right!" I see what you did there 😉

[@jakobr_]

Uhhhhh… yeah, that joke was totally intentional… I included it in the script on purpose… yeah… for sure…

[@Dr.1.]

LMAO now that you mention it... I get the joke absolutely genius

[@Dr.1.]

@@jakobr_ hats off

[@i_cam]

always thought of the law of cosines as the upgraded pythagorean theorem, but never seen the visual proof get upgraded as well. awesome stuff!

[@gudmundurjonsson4357]

this is amazing for being so simple, so directly connected to the standard proof for the pythagorean theorem, and somehow completely unknown. beautiful!

[@i_cam]

one phrase that i've been using recently to describe how higher levels of math/programming etc knowledge can help is that you are "ignoring less of the truth". Like the pythagorean theorem is just law of cosines but only the 90 degree case. That ab cos (C) term has been there all along, we just hadn't discovered it yet. This kinda logic comes up in a lot more places than you'd think, like the earth gravity equation (the 9.8m/s^2 one) being upgraded to the law of gravity (the one with big G), or like the whole chain of simplification from area of a quadrilateral -> area of a parallelogram -> to area of a rectangle -> area of a square. You can find a square's area using any of those formulas, we just use and learn the easiest and most obvious one, but as the math you do gets harder you have to uncover what's really been there all along. idk this video just really cemented for me how important that thought ive been having is.

[@BariScienceLab]

Hey! You should publish this in a math journal. This is great.

[@jakobr_]

Thank you! I’ve never really looked at publishing anything before, this is entirely new to me. Do you have any tips or recommendations?

[@BariScienceLab]

@@jakobr_ Sure, you can reach out to me at [e-mail] and I'll give some suggestions from my experience publishing papers. You can find my published pre-prints here: arxiv.org/search/?query=refath+bari&searchtype=all&source=header

[@happyhomosapien3138]

I'm a high school student, and this is how I like to learn the concepts. Thanks for sharing this video. This video is quite helpful. Keep making such videos. 👍

[@pauselab5569]

Well that’s not how math proofs work…

[@mszg1907]

Please, put more videos. We love your work! Thank you.

[@jamiepayton1574]

Really good stuff, I hope you get some recognition for this.

[@davidswitzer5743]

Wow! Great explanation and terrific animations. Keep up the good work Jake!

[@walkastray007]

People forget that math was created so we can represent the world. Equations arent just there, they represent something meaningful. Every equation and mathematical concept will have some form of visualization for why it's true, because that is simply what the equation and mathematical concept is representing.

[@harryfan8785]

Wooooooooooooooah.... there really is an intuitive explanation! Nice job, subbed!

[@Dr.1.]

ANIMATIONS ARE SOOO GOOOD I LOVE IT

[@matthewgiallourakis7645]

I believe Erdős would say "This one is from the book!"

[@jakobr_]

Wow, that is incredibly high praise! I definitely *felt* like I was reading from “the book” when I discovered it!

[@CognitiveOffense]

Truly delightful. Thank you.

[@LinesThatConnect]

Really cool proof!

[@themathsgeek8528]

This is an absolutely amazing proof, great video!!

[@Z7youtube]

This is absolutely great! i really liked the visual animations and the way you proved the law! such an underrated channel, the video popped up in my recommendations so i thought you're famous but i was surprised when i saw that this is your first video! keep up the great work!!

[@Xayuap]

you know, once I demostrate the cosines law, that day I realize pitagoras is abstracter, cause any triangle could be splited into two rectangle triangles. of course I had to use pitagoras in the cosines law demostration. Then, to demostrate pitagoras, there you got to be wise.

[@theodorostsilikis4025]

Mathologer had a great video for pythagorean theorem but i m not sure it still exists. All the geometry of this video was explaned there ( plus far more)

[@fahimmehtab7086]

Loved the video and the idea too.

[@richardbloemenkamp8532]

Very nice, great as an addition to the standard visual proof of Pythagoras.

[@kikivoorburg]

Amazing!

[@LunizIsGlacey]

Wow, very cute proof!

[@jannikhilger6739]

Great video

[@NPDGX]

Oh wow, this is spectacular!

[@minimath5882]

wish they showed this in school!

[@omardiaz6255]

Amazing, beautiful proof

[@marcfruchtman9473]

Super impressive. One has to wonder why isn't the Law of Cosines focused on more heavily that Pythagorean Theorem. Since it applies to all triangles, it seems to be the better version, with fewer inherent limitations.

[@terdragontra8900]

I think its cuz its just not quite as pretty looking

[@jakobr_]

I can think of a few reasons. The PT is “more efficient”, in a way, for certain very common practical applications: just three terms, one for each side. Doesn’t get simpler than that. It’s also easier to understand, since the LoC requires a little bit of trigonometry and understanding of how functions work. Generalization is cool, but it’s often less practical if there’s already a more specific tool for the job.

[@bladdnun3016]

It's basically the same reason as why manual screwdrivers are still in use when powered ones will do the job and typically come with a large set of bits.

[@fibbooo1123]

Great video!

[@KSJR1000]

This is excellent

[@jakobthomsen1595]

Beautiful!

[@thebees955]

Lovely stuff & great job with the animation! Have thought for some time that Pythagorean Theorem pales in comparison to the wonderful cosine rule!

[@jschnei3]

Very cool proof! I notice that you didn't animate the "sliding triangular blocks" demonstration of the correspondence of the two figures at 1:10 for the Py.Thm. It makes me wonder if there could be something analogous for your Loctagons.

[@jakobr_]

Sorry, I don’t think I understand what you’re referring to. I show something very similar to 1:10 at 2:40

[@jakobr_]

Actually I think I understand, you’re referring to the process of sliding the triangles within one figure to transform it into the other, right? Such a thing is definitely possible in the obtuse loctagons, it’s pretty easy to see. I can’t really picture it for the acute case though.

[@jschnei3]

@@jakobr_ Yes that is what I'm referring to, precisely. I'd like to see it for the obtuse loctagons.

[@Dr.1.]

THAT IS SOOOO SOO SOSOSOO COOLLLLL

[@lamenwatch1877]

I always knew it as "a^2 + b^2 = c^2" rather than "a^2 = b^2 + c^2" With 'c' being the hypotenuse rather than 'a' But I guess it can work either way.

[@Ghaith7702]

Well done

[@htvz]

Excelente prueba del Teorema del coseno

[@jakobr_]

Thank you! I'm glad that you understood it despite the English narration. The proof is in the pictures, that's all we need!

[@htvz]

@@jakobr_ It's math, language is universal :)

[@ShankarSivarajan]

Definitely from The Book.

[@omargaber3122]

❤❤❤

[@9sven6]

Very surprising that this one has so few views!

[@nuh-uh372]

With what did you make this animation? Its cool

[@jakobr_]

I used 3blue1brown’s Python library called Manim Community Edition. And thank you!

[@high4702]

Cool video! But could you left the link for the code

[@jakobr_]

I am not proud of the code I wrote for this video. It *technically* makes what I want it to, and that’s all it really has going for it… Inconsistent, inefficient, illegible, other “I” words, you get the idea, haha

[@Uuugggg]

The triangle you picked for the non-right triangle section... sure looks like a right triangle, just on the other side. The quiet parts like 2:16 are too quiet, too fast, not enunciated enough to understand

[@jakobr_]

Thanks for the feedback! Those triangles are very close to being right, but if you look closely, the green side is not exactly parallel to the red of the square. It was difficult to decide on a general-purpose example angle, and a compromise had to be made, in this case, appearing like another right triangle. But even if it was, it would still be a good demonstration of the Law of Cosines since the angle in question still is not right. The part at 2:16 is supposed to be just a side comment, it being quiet/fast was intentional. But I’ll keep that in mind next time I think about doing something like that again.

[@Neptoid]

This looks quite truncated 4:25

[@jakobr_]

Yep, that one does look a little strange, but that’s on purpose. I wanted to demonstrate that b and c are completely interchangeable with each other, even though I’d previously always just “declared” that c is the smaller side wlog. The result of c being larger is the same diagrams, but recolored, flipped and rotated a bit!

[@commentingonly4402]

What madman labels hypotenuse a??!!

[@jakobr_]

😎

[@kurihara9023]

hah lactocsa AhA DhA

[@keggluneq]

This was nice, but you really need to sloooow dooowwwnnn to 3 blue 1 brown speed. All your videos go by way too fast, and I have to incessantly pause and rewind. Very annoying.

[@jakobr_]

Yeah, the pacing is something I really gotta work on, thanks for the feedback. But… this is my first (and so far, only) video, are you thinking of someone else when you say “all your videos”?

[@keggluneq]

@@jakobr_ Sorry, there's another guy on YT who zips through his videos. My mistake. For a first video, this is excellent. Looking forward to future ones. IMO, 3 blue 1 brown has the perfect pace for this type of video.