Law Of Cosines II (visual proof)

This is a short, animated visual proof of the Law of Cosines using the Pythagorean theorem. This theorem relates the side of a triangle with the other two sides and the angle between those two sides.
If you like this video, consider subscribing to the channel or consider buying me a coffee: www.buymeacoff.... Thanks!
This animation is based on a proof from Timothy Sipka in the October 1988 issue of Mathematics Magazine (www.jstor.org/... ) page 259.
For another visual proof of this same fact check out:
• Law Of Cosines I (visu...
#mathshorts #mathvideo #math #trigonometry #lawofcosines #triangle #manim #animation #theorem #pww #proofwithoutwords #visualproof #proof #circle #pythagoreantheorem #obtuseangle #acuteangle #angle
To learn more about animating with manim, check out:
manim.community
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Music in this video:
Elegy by Asher Fulero

Nice little visual proof, but I think that the "algebraic manipulation" that change a sin into a cos would have been worth explaining. In fact, it feels like the heavy lifting of this proof is in the algebraic manipulation.

[@MathVisualProofs]

The sine doesn’t become a cosine. Sine squared plus cosine squared results in 1. The only algebra is the square of a difference.

@@MathVisualProofs I see. For me though, Sin square + Cos square equal 1 is more a Trig identity then just algebraic manipulation. Nonetheless, very nice visual proof. 🙂

[@MathVisualProofs]

@ Thanks! I do say "algebraic rules along with the pythagorean trigonometric identity" to indicate what needs to be done - so I agree that is more than algebraic manipulation.

[@brokenstone1670]

I felt this video was aptly brief and concise and had good amount of detail and that it holds up to the title of being a visual proof.

[@MathVisualProofs]

😀👍

[@dabullah]

Keep up these amazing videos!!

[@MathVisualProofs]

I’ll see what I can do. I’m over two years in and at some point I’ll run out of visual proofs I find interesting :)

[@RogatkaWR]

I love binging your videos! Keep it up!

[@MathVisualProofs]

Excellent! Thanks for watching :)

[@darthTwin6]

Beautiful proof. It really makes the whole thing so intuitive

[@MathVisualProofs]

Thanks!

[@michaelmurdoch]

Great video man! Loving this channel!

[@MathVisualProofs]

Glad to hear it! Thanks for watching!

[@techiesithastobetechies.8531]

Now this becomes hard to forget!

[@luukaskyronlahti5205]

This is awesome

[@MathVisualProofs]

Glad you like it!

[@woopnull9103]

2:08 why does b*sin(pi-theta) = b*sin(theta) ?

[@MathVisualProofs]

This is a fact about the sine function, which measures the y-coordinate on the unit circle after rotating angle theta. So if you instead rotate pi-theta you have the same y-coordinate.

[@TheEGod.]

This was confusing since I thought of it like theta minus pi, and not like pi minus theta.

[@MathVisualProofs]

@@TheEGod. theta-pi doesn’t give you the right identity.

[@TheEGod.]

@@MathVisualProofs Yeah thats why I said I was so confused. Since I was thinking of something inccorect.

[@MathVisualProofs]

@@TheEGod. Oh! I see. :)

[@pbworld7858]

I prefer this proof to the previous one.

[@sdspivey]

You should have pointed out the sin²+cos²=1. Many viewers may not know trig identities.

[@MathVisualProofs]

I do mention you need the "pythagorean trigonometric identity" so that people can search that if they aren't aware.

[@Botiful-Chaos]

In the obtuse example was side a for all the base or soecific part of it ?

[@ekoi1995]

We are given the equation: c^2 = (a - b * cos(t))^2 + (b * sin(t))^2 We aim to show that this equation simplifies to: a^2 + b^2 = c^2 + 2 * a * b * cos(t) Expanding the terms First, expand the terms on the right-hand side of the equation: (a - b * cos(t))^2 = a^2 - 2 * a * b * cos(t) + b^2 * cos^2(t) (b * sin(t))^2 = b^2 * sin^2(t) Now substitute these expanded forms into the original equation: c^2 = (a^2 - 2 * a * b * cos(t) + b^2 * cos^2(t)) + b^2 * sin^2(t) Simplifying the equation Combine the terms: c^2 = a^2 - 2 * a * b * cos(t) + b^2 * (cos^2(t) + sin^2(t)) Recall the Pythagorean identity: cos^2(t) + sin^2(t) = 1 So, substitute this identity into the equation: c^2 = a^2 - 2 * a * b * cos(t) + b^2 Rearranging the terms Now, let's rewrite the equation to match the desired form. Rearranging the terms: c^2 = a^2 + b^2 - 2 * a * b * cos(t) Finally, add 2 * a * b * cos(t) to both sides to get: a^2 + b^2 = c^2 + 2 * a * b * cos(t) Conclusion Thus, we have shown that: c^2 = (a - b * cos(t))^2 + (b * sin(t))^2 leads to the equation: a^2 + b^2 = c^2 + 2 * a * b * cos(t)

[@Botiful-Chaos]

And it is weird because in your second you added a right triangle and proved it like that i mean it does not make sense because you are solving for a different triangle now not an obtuse triangle so how both are related?

[@StratosFair]

Beautiful. What do you mean when you say this is equivalent to Pythogarean theorem though ? If we didn't have it in the first place we wouldn't be able to prove the law of cosines, no ?

[@MathVisualProofs]

You can prove the law of cosines without the Pythagorean theorem.

[@abhayphotos4398]

Amazing❤

[@MathVisualProofs]

Thank you!

[@GourangaPL]

why is Pitagorean Theorem called by the mathematician's name and i almost never see this one called Al-Kashi's law of cosines?

[@MathVisualProofs]

Yes, PT should be called "Right triangle theorem". Unfortunately it is how it is.

[@rttt284]

I really liked your work and explanation. Can you help me with the name of the program you are using for the explanation

[@MathVisualProofs]

I use manim for this.

[@VictorHarrys]

It’s not a fully “visual proof” since you included “algebraic manipulation”.

[@monoman4083]

very good....

[@MathVisualProofs]

Thanks!

[@Harshit-b2y]

I have doubt that if we have values of trigonometric ratios grater 90⁰, then why we can't use them directly?

[@gerardsagliocca6292]

Kind of speeded through. If you slow down a bit and added a few more algebraic steps, then the video would be great.

[@MathVisualProofs]

Noted. Thanks.

[@MuhammadAnas_Official]

Love it❤❤❤

[@MathVisualProofs]

Thanks!

[@yolomc2]

2:09 b*sin(pi-theta) = bsin(theta) This makes absolutely no sense

[@MathVisualProofs]

Do you know what the sine function does? It measures the y-coordinate on unit circle. If you rotate theta degrees you will be at the same y-coordinate as if your rotate theta degrees back from a pi rotation.

[@yolomc2]

@@MathVisualProofs Ok i get it now, I think i got confused when trying to picture the obtuse triangle inside a circle and the epicentre (theta) not being in the middle of the imaginary circle 😅 👍

[@WahranRai]

You must cote Al Kashi for this theorem as you cote Pythagore !

[@HolloMatlala1]

One day the importance and need of VR and Augmented Reality will be realized in Schools and Universities

[@수하긴]

우와..........

[@MathVisualProofs]

👍😀

[@asparkdeity8717]

❤️

[@loooooooon3666]

I still don't understand why it's Pi minus the angle alpha

[@MathVisualProofs]

That is the supplementary angle. Two angles that create a straight line must add to 180 degrees or pi rads

[@JakirHossain-ik5rp]

Software name please?

[@MathVisualProofs]

Manim. It’s in the description of every video.

[@electrozito9718]

It is not called the law of cosines , it is called Alkashi's theory originally made by one of the greatest arabic mathematicians. I think we should give the honor to his name and not change his theory's name.

[@friedrichhayek4862]

Irrelevant, and neither is a theory.

[@electrozito9718]

@@friedrichhayek4862 i meant Theorem but u got the point

[@rhandhom1]

I understand some of these words.

[@MathVisualProofs]

Start with those and branch out to the others 👍😀

[@sunnoah6502]

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[@the69paradise68]

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[@richarda6813]

A little too rushed

[@MathVisualProofs]

👍

[@RegularKerico]

It's a nice video, but calling it a visual proof seems like a stretch.

[@MathVisualProofs]

I mean, it is a published "proof without words." So in my mind, it's a pretty good visual proof. I added some commentary, but I think it stands alone too.