# Hyperbolic Sine Function is Odd

## Theorem

Let $\sinh: \C \to \C$ be the hyperbolic sine function on the set of complex numbers.

Then $\sinh$ is odd:

$\map \sinh {-x} = -\sinh x$

## Proof 1

 $\displaystyle \map \sinh {-x}$ $=$ $\displaystyle \frac {e^{-x} - e^{-\paren {-x} } } 2$ Definition of Hyperbolic Sine $\displaystyle$ $=$ $\displaystyle \frac {e^{-x} - e^x} 2$ $\displaystyle$ $=$ $\displaystyle -\frac {e^x - e^{-x} } 2$ $\displaystyle$ $=$ $\displaystyle -\sinh x$

$\blacksquare$

## Proof 2

 $\displaystyle \map \sinh {-x}$ $=$ $\displaystyle -i \, \map \sin {-i x}$ Hyperbolic Sine in terms of Sine $\displaystyle$ $=$ $\displaystyle i \, \map \sin {i x}$ Sine Function is Odd $\displaystyle$ $=$ $\displaystyle -\sinh x$ Hyperbolic Sine in terms of Sine

$\blacksquare$