Real 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:
 * $\Re \paren {\sin z} = \sin x \cosh y$

where:
 * $\Re z$ denotes the real part of a complex number $z$
 * $\sin$ denotes the sine function (real and complex)
 * $\cosh$ denotes the hyperbolic cosine 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 real part of a complex number.