A question over the reality of $\sin x$ 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?
 A: 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...
A: 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
$$\mathfrak{Re}\{e^{ix}\} = \cos x = \frac{e^{ix} + e^{-ix}}{2}$$
