# Inverse Hyperbolic Sine is Odd Function

## Theorem

Let $x \in \R$.

Then:

$\map {\sinh^{-1} } {-x} = -\sinh^{-1} x$

where $\map {\sinh^{-1} } {-x}$ denotes the inverse hyperbolic sine function.

## Proof

 $\displaystyle \map {\sinh^{-1} } {-x}$ $=$ $\displaystyle y$ $\quad$ $\quad$ $\displaystyle \leadstoandfrom \ \$ $\displaystyle -x$ $=$ $\displaystyle \sinh y$ $\quad$ Definition 1 of Inverse Hyperbolic Sine $\quad$ $\displaystyle \leadstoandfrom \ \$ $\displaystyle x$ $=$ $\displaystyle -\sinh y$ $\quad$ $\quad$ $\displaystyle \leadstoandfrom \ \$ $\displaystyle x$ $=$ $\displaystyle \map \sinh {-y}$ $\quad$ Hyperbolic Sine Function is Odd $\quad$ $\displaystyle \leadstoandfrom \ \$ $\displaystyle \sinh^{-1} x$ $=$ $\displaystyle -y$ $\quad$ Definition 1 of Inverse Hyperbolic Sine $\quad$ $\displaystyle \leadstoandfrom \ \$ $\displaystyle -\sinh^{-1} x$ $=$ $\displaystyle y$ $\quad$ $\quad$

$\blacksquare$