# Inverse Hyperbolic Sine of Imaginary Number

## Theorem

$\sinh^{-1} \left({i x}\right) = i \sin^{-1} x$

## Proof

 $\displaystyle y$ $=$ $\displaystyle \sinh^{-1} \left({i x}\right)$ $\displaystyle \implies \ \$ $\displaystyle \sinh y$ $=$ $\displaystyle i x$ Definition of Inverse Hyperbolic Sine $\displaystyle \implies \ \$ $\displaystyle i \sinh y$ $=$ $\displaystyle - x$ $i^2 = -1$ $\displaystyle \implies \ \$ $\displaystyle \sin \left({i y}\right)$ $=$ $\displaystyle - x$ Hyperbolic Sine in terms of Sine $\displaystyle \implies \ \$ $\displaystyle i y$ $=$ $\displaystyle \sin^{-1} \left({- x}\right)$ Definition of Complex Inverse Sine $\displaystyle \implies \ \$ $\displaystyle i y$ $=$ $\displaystyle -\sin^{-1} x$ Inverse Sine is Odd Function $\displaystyle \implies \ \$ $\displaystyle y$ $=$ $\displaystyle i \sinh^{-1} x$ multiplying both sides by $-i$

$\blacksquare$