Solution
= Solution
Expand the <sine> of the <complex number> using <hyperbolic functions>:
$$
\sin(x+iy)=\sin x\cosh y+i\cos x\sinh y.
$$
Its squared <complex modulus> is
$$
|\sin(x+iy)|^2=\sin^2x\cosh^2y+\cos^2x\sinh^2y=\sin^2x+\sinh^2y\geq\sin^2x.
$$
Taking nonnegative <square roots> proves $\boxed{|\sin z|\geq|\sin x|}$. Equality holds exactly when $y=0$.