Imaginary Part of Sine of Complex Number

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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.

$\blacksquare$


Sources