# Inverse Sine of Imaginary Number

## Theorem

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

## Proof

 $\ds y$ $=$ $\ds \map {\sin^{-1} } {i x}$ $\ds \leadsto \ \$ $\ds \sin y$ $=$ $\ds i x$ Definition of Complex Inverse Sine $\ds \leadsto \ \$ $\ds i \sin y$ $=$ $\ds -x$ $i^2 = -1$ $\ds \leadsto \ \$ $\ds \map {\sin^{-1} } {i y}$ $=$ $\ds -x$ Sine in terms of Hyperbolic Sine $\ds \leadsto \ \$ $\ds i y$ $=$ $\ds \map {\sinh^{-1} } {-x}$ Definition of Inverse Hyperbolic Sine $\ds \leadsto \ \$ $\ds i y$ $=$ $\ds -\sinh^{-1} x$ Inverse Hyperbolic Sine is Odd Function $\ds \leadsto \ \$ $\ds y$ $=$ $\ds i \sinh^{-1} x$ multiplying both sides by $-i$

$\blacksquare$