Question

Why does the taylor series of e^x line up with sine and cosine's at all?

Answer 1

Any guesses? I just can't help but feel like a victim to his miraculous coincidence.

Answer 2

line up ?

Answer 3

Sure, as in e^ix=cosx+isinx. That's just lucky.

Answer 4

you know sin x and cos x can be expressed in the form e^ix and e^-ix right ?

Answer 5

Yeah, that's what I mean, why?

Answer 6

now what do u mean by line up? :P

Answer 7

Why should their taylor series be so similar that just plugging in sqrt(-1) allows us to equate them. It's unreasonably lucky.

Answer 8

\[e^x=1+x+\frac{ x^2 }{ 2! }+\frac{ x^3 }{ 3! }+\frac{ x^4 }{ 4! }+\frac{ x^5 }{ 5! }+...\]\[\sin(x)=x-\frac{ x^3 }{ 3! }+\frac{ x^5 }{ 5! }-...\]\[\cos(x)=1-\frac{ x^2 }{ 2! }+\frac{ x^4 }{ 4! }-...\]

Now it's easy to see how plugging in e^(ix) is equal to cosx+isinx. This is the only reason I believe this formula is true, since I can very reliably do this myself.

But I see no reason as to why this should be true. I'm comfortable with using this formula even, I've been using it for probably a year or more and it's incredibly convenient. But I just don't understand why this would ever be possible.

Answer 9

you can get series of e^x from series of sin and cos using the relation e^ix = cos x +isin x

similarly, you can get series of sin x or cos x, using their e^ix formulas...

Answer 10

you know formula is true but you don't know why is it true?

Answer 11

Yeah you're not really following me I think. Why would there be such a relationship in the first place? Algebraically it makes perfect sense. The taylor series when graphed match the graph of what they should.

But if I never saw a taylor series representation, there would never be a reason for me to believe e^ix had anything to do with sine or cosine. Do you see where I'm coming from?

Answer 12

I know why it's true, but I don't know why it should be true.

Answer 13

now my head hurts :P
i'll leave this for other experts....

Answer 14

If some other random number that wasn't e when raised to (ix) power had the same taylor series as cosx+isinx then I'd believe that too. It just sort of doesn't make sense to me at all, I'm a victim to "e" for almost no good reason haha.

Answer 15

at this point your speculation is as good as mine haha.

Answer 16

they are no so similar,
e^x will go on increasing (all terms are positive) when x>0 and go on decreasing when x<0
imagine e^x and e^-x curves.

whereas sin and cos has alternate + and - signs, so they increase , decrease, increase, decrease, just like their behavior...

Answer 17

Yeah, exactly. It's FREAKIN WEIRD.

Answer 18

A philosophical discussion? How was e originally defined anyway?