Math's most beautiful identity
ฝัง
- เผยแพร่เมื่อ 4 พ.ย. 2024
- In this video I teach about some basics of trigonometry, and one of my favorite identities from trig.
This video was made as a submission to the Summer of Math Exposition 2 (#SoME2) constest. I animated the video using a mix of a custom JS library I wrote for the graphics, as well as 3blue1brown's Manim for the expressions and text.
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Twitter: / zenzicubic
Website: zenzicubic.dev
Music by Vincent Rubinetti
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Huh, I never thought about it that way-very cool-I appreciate the compact delivery. I hope you make lots more maths videos!!
Thanks, and yes I plan to make more :)
Really appreciate it
I just leant some trigonometric identites in my school . Looking at visualisation makes it more clear . We need more of this content :D
You're going to be the next big math channel, and I can't wait.
This is one of those identities that are beautiful less because they are true, and more because they can't be anything else. x^2 + y^2 = d^2 is the fundamental equation for distance, and a circle can be described as x = rcos(t), y = rsin(t), but a circle is also described as the set of all points a certain distance from its center x^2 + y^2 = r^2, which is just the distance equation from earlier, but with r as a constant. Plug in the parametric form and you get r^2cos^2(t) + r^2sin^2(t) = r^2. Divide out the r^2 so the radius is one and you're left with cos^2(t) + sin^2(t) = 1. There's nothing else it _could_ be.
Or... at least there isn't when distance is defined as x^2 + y^2 = d^2, which is the case in standard euclidean geometry. There are however other ways of defining distance that yield different versions of cos^2(t) + sin^2(t) = 1. One example is if x^2 *-* y^2 = d^2, which seems weird, but it's completely valid and very useful for talking about hyperbolic geometry, which includes spacetime. Here, set of points of a given radius r can be described as x^2 - y^2 = r^2. What's the parametric form? Well it's just like the circular case, but modified slightly: x = rcosh(t), y = rsinh(t). Put these together and you get a different identity: cosh^2(t) - sinh^2(t) = 1. (And then there are spaces where you change the exponent from 2 to something else, which gives its own versions of sine and cosine.)
That's interesting, I never thought that hyperbolic geometry has its own version of the identity. I know about hyperbolic trig though...but never made much use of it.
@@Zenzicubic Among the beautiful connections between circular geometry and hyperbolic geometry is through complex numbers. Namely: sinh(ix) = isin(x) and cosh(ix) = cos(x). Euler's formula also extends since e^ix = cos(x) + isin(x) is effectively a special case of e^x = cosh(x) + sinh(x). Specially cosh and sinh are the anti-symmetric (odd) part and symmetric (even) part of the exponential respectively.
cosh(x) = (e^x - e^(-x))/2
cos(x) = cosh(ix) = (e^ix - e^(-ix))/2
sinh(x) = (e^x + e^(-x))/2
sin(x) = sinh(ix) = (e^ix + e^(-ix))/2
e^ix = cosh(ix) + sinh(ix) = cos(x) + isin(x)
Complex numbers are effectively the fundamental numbers of elliptical geometry. Hyperbolic geometry has the split-complex numbers with j^2 = 1, leading to e^jx = cosh(x) + jsinh(x). Note that neither really corresponds to flat geometry. What might be a number system in between i^2 = -1 and j^2 = 1? ;) Naturally, this number system would also have its own version of Euler's formula and the distance equation.
Probably on the top 10. I prefer a few others but the sum of the squares of sine and cosine is very high on the list.
I agree, I have a few favorites too, but this one is my favorite by far.
Such a good video!
Thanks for your nice comments :))
how to make suck great animations
In replay to another comment, the creator says they use a custom java script library which they have tutorial videos of on their channel-as well as a little bit of Maniam
cool
Wow😁so happy that i found this channel
Nice work with the visuals! However I believe that at the end of the video you are not adding anything new, since you use the Pythagorean Theorem to "prove" itself.
Yes, you could think of it that way, but I usually think of them as separate mathematical entities. I'm glad you enjoyed the video btw.
Like it! :)
Thanks, this is the first time I've done this kind of thing. Glad people are enjoying it :))
Really nice video, is this the same python lib that 3b1b uses for his animations?
Partially, the text animations are done with that and the rest is done with a custom Javascript library I wrote
I don't understand this part 1:35
Why is cosine of phi the x axis
And sine of of phi the y axis
Whats the reason behind it?
Well, that just happens to be what they're named. I have no clue where the words "sine" and "cosine" came from, they just sort of are. Sorry if I made that unclear.
These are names given by convention, there isn't much more to it than that. Cosine is what we call the projection of the point onto the x-axis and sine is what we call the projection onto the y-axis. And if you wonder about the origins of the words, then don't expect to find something reasonable. Basically, the term sine comes from a latin translation of an arabic translation of the original hindi word for this concept, but it was unfortunately mistranslated to the latin word for "stomach". The term cosine comes from thinking of cosine as the "complement of" sine, hence "co"sine.
@@Zenzicubic no i wasn't asking about the origin of their names, who cares about them anyways.
I was asking why the cos is the x co ordinate of the point, and sin is the y co ordinate of the point.
Using Manim?
Partially, the animated text and expressions are done with Manim, while the diagrams and graphs are done using my custom Javascript library. I have some other demo of it on my channel