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How to prove Euler’s formula: $\exp(i t)=\cos(t)+i\sin(t)$?

I need to know why Euler's formula is true? I mean why is the following true: $$ e^{ix} = \cos(x) + i\sin(x) $$

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IremadzeArchil19910311
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    Wrong name, wrong formula. – Did Oct 31 '12 at 07:31
  • You can try plugging $i \theta$ into the Taylor series for $e^x$, and watch as the Taylor series for $\sin$ and $\cos$ appear, as if by magic. I imagine it was quite a shock to Euler when he did this. – littleO Oct 31 '12 at 07:31
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    $\frac{d}{dx} e^{ix}=ie^{ix}\ \frac{d}{dx} \cos{x}+i\sin{x}=i(\cos{x}+i\sin{x})$ – Angela Pretorius Oct 31 '12 at 07:37
  • See: http://pundit.pratt.duke.edu/wiki/Complex_Numbers and http://en.wikipedia.org/wiki/Complex_number – NoChance Oct 31 '12 at 10:32

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Hint: Notice $$\sin (x) = x - \frac{x^3}{3!} + \frac{x^5}{5!} - \frac{x^7}{7!} + ..... $$ and $$i\cos (x) = i - i\frac{x^2}{2!} + i\frac{x^4}{4!} - i\frac{x^6}{6!} + .... $$ Now add them and use the fact that $i^2 = -1$, $i^3 = -i$, $i^4 = 1$. You should obtain $e^{ix}$. Also notice: $$e^x = 1 + x + \frac{x^2}{2!} + \frac{x^3}{3!} + ....... $$

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