Yup, the proof actually uses Taylor series expansions.
Just substitute i*theta for x in these sums and note some theorems about adding and subtracting absolutely convergent sums.

>>226

That method uses the derivatives of sin and cos. Is it possible to derive them without pythagoras aswell?

>227
Well its hard to define sin and cos at all without pythagoras, I think one of the only ways would be to say that the sum IS the sin function.
I.E. say that sinx=(that sum) by definition, and THEN show its geometric properties.

>>230
>Well its hard to define sin and cos at all without pythagoras,

Nah, sin and cos are just the ratios of the sides of a traingle. The pythagoras theorem shows how they are related but it's not required for their definition.

>>233
Not that I'm disagreeing,I've actually never seen anyone get from the geometric interpretation to Euler's formula if you can or have a link somewhere please share. I would really appreciate it.

It is possible, I did it in calculus II, using the Power Series expansions. You can also use the limit definition, and polar coordinates. It shouldn't be too hard to find these proofs floating around on the web.

>>227
actually now that i think about it, you don't need to know anything about derivatives. All you need to do is show that sin is indeed equal to that sum.

>>227
Yes, it is possible to find the derivatives of sine and cosine without the Pythagorean theorem. In fact, the only trigonometric identities required are the angle addition formulas, which do not require the Pythagorean theorem for their derivations either. Curiously, the Pythagorean theorem can be derived from Euler's formula:

http://www.mandysnotes.com/Trigonometry-With-Complex-Numbers/Complex-Trigonometry/The-Pythagorean-Theorem-from-Eulers-Formula

The writer of the derivation I linked to messed up and threw in a + when there should have been a -, but you get the idea. Anywho, what I'm getting at is that Euler's formula actually cannot be derived from the Pythagorean theorem, but it is the other way around.

>>242

Yeah, that's why i wanted it.

http://www.boards.ie/vbulletin/showthread.php?t=2056785182

>>242
Thanks!