How do we know that the Sine function has no Non-Real Roots?

$$\sin(x+iy)=\sin x\cosh y+i\cos x\sinh y$$ Note that $\cosh y$ is never zero for $y$ a real number. Thus, if $\sin(x+iy)=0$, $\sin x=0$ (the real and imaginary parts must be identically zero). But this forces $\cos x=\pm1$, which in turn forces $\sinh y=0$ and thus $y=0$. All the roots must be real.

Since $\sin z = \frac{1}{2i} ( e^{iz} - e^{-iz} )$, we see that $\sin z = 0$ if and only if $e^{iz} = e^{-iz}$, or $e^{2iz}=1$. Taking the modulus of both sides, we see that $$ 1 = |1| = |e^{2iz}| = e^{\Re(2iz)} = e^{-2\mathop{\rm Im} z}, $$ which implies that $-2\mathop{\rm Im} z=0$ and thus $\mathop{\rm Im} z=0$.