Real 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:

$\map \Re {\sin z} = \sin x \cosh y$

where:

$\map \Re z$ 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:

$\map \sin {x + i y} = \sin x \cosh y + i \cos x \sinh y$

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

$\blacksquare$


Sources