Modulus of Sine of Complex Number

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

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


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


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


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