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.

For another visual proof of this same fact check out:

#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:
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Music in this video:
Elegy by Asher Fulero

[ekoi]

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)

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.

[brokenstone]

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.

[RogatkaWR]

I love binging your videos! Keep it up!

[darthTwin]

Beautiful proof. It really makes the whole thing so intuitive

[pbworld]

I prefer this proof to the previous one.

[woopnull]

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

[Ibrahim_Ezzeddine-dy]

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?

[Ibrahim_Ezzeddine-dy]

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

[Harshit-by]

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

[sdspivey]

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

[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 ?

[VictorHarrys]

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

[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?

[rttt]

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

[gerardsagliocca]

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

[Squidofterror]

I'm so sad i thought I created the second one for a while although I did "discover" it in a different way

[HolloMatlala]

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

[WahranRai]

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

[yolomc]

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