Imaginary Part of Sine of Complex Number

Theorem
Let $z = x + i y \in \C$ be a complex number, where $x, y \in \R$.

Let $\sin z$ denote the complex sine function.

Then:
 * $\Im \paren {\sin z} = \cos x \sinh y$

where:
 * $\Im z$ denotes the imaginary part of a complex number $z$
 * $\sin$ denotes the complex sine function
 * $\cos$ denotes the real cosine function
 * $\sinh$ denotes the hyperbolic sine function.

Proof
From Sine of Complex Number:
 * $\sin \paren {x + i y} = \sin x \cosh y + i \cos x \sinh y$

The result follows by definition of the imaginary part of a complex number.