Originally Posted by **TD!**

You can use that identity of course, but I was referring to the definition:

$\displaystyle \sin z \equiv \frac{{e^{iz} - e^{ - iz} }}{{2i}}$

The exponential function is periodic, with period $\displaystyle 2\pi i$. We can now use this to find the zeroes.

$\displaystyle

\sin z = 0 \Leftrightarrow \frac{{e^{iz} - e^{ - iz} }}{{2i}} = 0 \Leftrightarrow e^{iz} - e^{ - iz} = 0 \Leftrightarrow e^{iz} = e^{ - iz}

$

Now we use the periodicy of e^z.

$\displaystyle

e^{iz} = e^{ - iz} \Leftrightarrow iz = - iz + 2k\pi i \Leftrightarrow 2zi = 2k\pi i \Leftrightarrow z = k\pi

$