# Hyperbolic Sine in terms of Sine

## Theorem

Let $z \in \C$ be a complex number.

Then:

$i \sinh z = \map \sin {i z}$

where:

$\sin$ denotes the complex sine
$\sinh$ denotes the hyperbolic sine
$i$ is the imaginary unit: $i^2 = -1$.

## Proof

 $\displaystyle \map \sin {i z}$ $=$ $\displaystyle \frac {e^{i \paren {i z} } - e^{i \paren {-i z} } } {2 i}$ Sine Exponential Formulation $\displaystyle$ $=$ $\displaystyle \paren {-i} \frac {e^{-z} - e^z} 2$ $i^2 = -1$ $\displaystyle$ $=$ $\displaystyle i \frac {e^z - e^{-z} } 2$ $i^2 = -1$ $\displaystyle$ $=$ $\displaystyle i \sinh z$ Definition of Hyperbolic Sine

$\blacksquare$