# Modulus 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:

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

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 $\displaystyle$ $=$ $\displaystyle \sin^2 x \paren {1 + \sinh^2 y} + \cos^2 x \sinh^2 y$ 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 \ \$ $\displaystyle \cmod {\sin z}$ $=$ $\displaystyle \sqrt {\sin^2 x + \sinh^2 y}$

$\blacksquare$