Modulus 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:

$\cmod {\sin z} = \sqrt {\sin^2 x \sinh^2 y}$

where:

$\cmod z$ denotes the modulus of a complex number $z$
$\sin x$ denotes the real sine function
$\sinh$ denotes the hyperbolic sine function.


Proof

\(\displaystyle \sin \paren {x + i y}\) \(=\) \(\displaystyle \sin x \cosh y + i \cos x \sinh y\) Sine of Complex Number
\(\displaystyle \leadsto \ \ \) \(\displaystyle \cmod {\sin z}^2\) \(=\) \(\displaystyle \paren {\sin x \cosh y}^2 + \paren {\cos x \sinh y}^2\) Definition of Complex Modulus
\(\displaystyle \) \(=\) \(\displaystyle \sin^2 x \paren {1 + \sinh y}^2 + \cos^2 x \sinh^2 y\) Difference of Squares of Hyperbolic Cosine and Sine
\(\displaystyle \) \(=\) \(\displaystyle \sin^2 x + \paren {\sin^2 x + \cos^2 x} \sinh^2 y\) rearranging
\(\displaystyle \) \(=\) \(\displaystyle \sin^2 x + \sinh^2 y\) Sum of Squares of Sine and Cosine
\(\displaystyle \leadsto \ \ \) \(\displaystyle \cmod {\sin z}\) \(=\) \(\displaystyle \sqrt {\sin^2 x + \sinh^2 y}\)

$\blacksquare$


Sources