By Euler's formula, I can express $i$ in the following way: $$ i = \cos(\pi/2) + i \sin(\pi/2) = \exp(i \pi/2) $$ I wonder if it is legitimate to write

$$\begin{align}i \sin x &= \exp(i\pi/2) \cdot \frac{\exp(ix)-\exp(-ix)}{2i} \\ &= \frac{\exp(ix)\exp(i\pi/2)-\exp(-ix)\exp(i\pi/2)}{2i} \\ &= \frac{\exp(i(x+\pi/2))-\exp(-i(x+\pi/2))}{2i} \\ &= \sin(x+\pi/2) \end{align}$$

I don't feel like this is right, because it would imply a lot of weird things. So where is my mistake? note: I'm really missing sleep, so please be kind with me.


In your sleep-deprived delirium, you have said $$ e^{-ix}e^{i\pi/2} = e^{-i(x+\pi/2)}$$ where it is actually $$ e^{-ix}e^{i\pi/2} = e^{-i(x-\pi/2)}.$$

It would certainly be strange if $i\sin(x) = \sin(x+\pi/2)$ since one is pure imaginary and the other real for real $x.$

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    $\begingroup$ Thank you, I shouldn't even be trying. I will drown in self-pity till I fall asleep. $\endgroup$ – Marc Oct 26 '17 at 23:30

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