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Harmonic functions are in widespread use in physical descriptions of natural real phenomena. I am just wondering therefore how we can define $\sin(x)$ to be part of a real physical quantity (with regard to some plane wave or electromagnetic theory, I'm not a physicist so please insert relevant theory here!) if, as shown by Euler it can be expressed as a complex exponential?

\begin{equation} \sin \theta =\frac {e^{i\theta}-e^{-i\theta}}{2i} \end{equation}

My intuition would say that we always have to describe observables in any sense as a square of a sine, but how about a wave packet of this form?

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The expression you have for $\sin$ is real, as can be understood by the following:

$$\begin{align}e^{i\theta}&=\cos\theta+i\sin\theta\\ \text{and so}\\ \frac{e^{i\theta}-e^{-i\theta}}{2i}&=\frac{\cos\theta+i\sin\theta - \cos\theta+i\sin\theta}{2i}\\ &=\sin\theta\end{align}$$

There is no complex number when you're done expanding...

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Note that the numerator is the difference of a complex number and its conjugate. This difference is imaginary (or zero) for any complex number:

$$z - z^* = 2i\mathfrak{Im}\{z\}$$

To see this, write $z = \sigma + i\omega$. Then

$$z - z^* = (\sigma + i\omega) - (\sigma - i\omega) = 2i\omega$$

Thus, if we want the imaginary part of a complex number (which is real), we have

$$\mathfrak{Im}\{z\} = \omega = \frac{z - z^*}{2i}$$

Then, since by Euler's formula we have

$$e^{ix}= \cos x + i\sin x $$

it follows that

$$\mathfrak{Im}\{e^{ix}\} = \sin x = \frac{e^{ix} - e^{-ix}}{2i}$$

Finally, if want the real part of a complex number, we have

$$\mathfrak{Re}\{z\} = \sigma = \frac{z + z^*}{2}$$

thus

$$\mathfrak{Re}\{e^{ix}\} = \cos x = \frac{e^{ix} + e^{-ix}}{2}$$

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