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You are not logged in. #1 20060319 10:43:30
Euler's forumulaIt is a pretty cool thing: One professor of mine made a pretty bold claim the other day about it. He said that any trig identity can be derived using this and only this formula. So here's an idea. Take a trig identity and see if you can prove it using nothing else. If you can, post it here. Any identity that we can prove using only Euler's Formula is fair game (i.e. you can use it in another proof), but you gotta prove it first before you can use it. If you run into one that you have trouble with, by all means, post it. I'm a bit short on time, so I'll post one up later tonight. Edit: Whoops, thanks mathsyperson Last edited by Ricky (20060320 03:24:09) "In the real world, this would be a problem. But in mathematics, we can just define a place where this problem doesn't exist. So we'll go ahead and do that now..." #2 20060320 01:08:57
Re: Euler's forumulaShould that be ?I'm not that good with complex numbers, so I'm not sure how useful I'll be, but I'll give it a go later on, when I've got some more free time. Why did the vector cross the road? It wanted to be normal. #3 20060320 03:23:47
Re: Euler's forumulaThe only real property of imaginary numbers you really have to know is that: "In the real world, this would be a problem. But in mathematics, we can just define a place where this problem doesn't exist. So we'll go ahead and do that now..." #4 20060320 03:26:49
Re: Euler's forumulaThere is another property that is hidden in there if you can find it. Remember: "In the real world, this would be a problem. But in mathematics, we can just define a place where this problem doesn't exist. So we'll go ahead and do that now..." #5 20060320 03:51:01
Re: Euler's forumulaYes, I noticed that. OK, I'll shamelessly steal your LaTeX and alter it for my own purposes then. Why did the vector cross the road? It wanted to be normal. #6 20060320 04:08:16
Re: Euler's forumulaNicely done. Now I suggest we go after cos²θ + sin²θ = 1, as that will most likely be a useful property to use for other identites later on. "In the real world, this would be a problem. But in mathematics, we can just define a place where this problem doesn't exist. So we'll go ahead and do that now..." #7 20060320 13:56:48
Re: Euler's forumulae^ix e^ix = (cosx+isinx)(cosxisinx) < sin(x) =sinx > = (cos^2[x]  i^2 sin^2[x])=(cos[x])^2+(sin[x])^2 X'(yXβ)=0 #8 20060320 14:57:25
Re: Euler's forumulaAbsolutely right, George Y. By Abraham de Moivre's theorem, (Cosx+iSinx)(CosxiSinx)=Cos²x  (i²)Sin²x = Cos²x  (1)Sin²x = Cos²x + Sin²x Therefore, Cos²x + Sin²x=1. Character is who you are when no one is looking. #9 20060320 15:32:43
Re: Euler's forumulaOne property used in that proof is cos(x) = cos(x) and sin(x) = sin(x). I have yet to find a way to prove these using e^iθ, but for the moment, do so using the definitions of sine and cosine. Last edited by Ricky (20060320 15:47:25) "In the real world, this would be a problem. But in mathematics, we can just define a place where this problem doesn't exist. So we'll go ahead and do that now..." #10 20060322 12:22:10
Re: Euler's forumulaActually, let's skip cos(x) = cos(x) and sin(x) = sin(x) for now. I believe they will come up soon enough. "In the real world, this would be a problem. But in mathematics, we can just define a place where this problem doesn't exist. So we'll go ahead and do that now..." #11 20060324 02:21:31
Re: Euler's forumulaCircular proof, i am sorry to say X'(yXβ)=0 #12 20060324 02:22:44
Re: Euler's forumulaThough a very good angle to memorize those formulas X'(yXβ)=0 #13 20060324 08:13:19
Re: Euler's forumulaAnd why exactly is it circular, George? "In the real world, this would be a problem. But in mathematics, we can just define a place where this problem doesn't exist. So we'll go ahead and do that now..." #14 20060324 12:31:53
Re: Euler's forumulabecause in order to prove Euler's formula, we commonly use Taylor series. Last edited by George,Y (20060324 12:33:46) X'(yXβ)=0 #15 20060324 15:40:49
Re: Euler's forumulaGeorge, I agree. And actually, I agreed before you ever backed up your statement. I really just wanted you to start showing what you said instead of just stating it. "In the real world, this would be a problem. But in mathematics, we can just define a place where this problem doesn't exist. So we'll go ahead and do that now..." #16 20060325 00:21:01
Re: Euler's forumulaNever mind. we just have different angles X'(yXβ)=0 #17 20060408 16:09:44
Re: Euler's forumulaIt really has an great application!! ... last part ,+ uncertain, when n is even, or when n is odd Last edited by George,Y (20060408 16:55:10) X'(yXβ)=0 #18 20060410 02:24:17
Re: Euler's forumulacos(180°)= X'(yXβ)=0 #19 20060410 11:05:15
Re: Euler's forumulap²10p(1p)+5(1p)²=0 corrected Last edited by George,Y (20060410 11:06:15) X'(yXβ)=0 