Inverse Hyperbolic Sine is Odd Function

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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

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

$\blacksquare$


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