Inverse Sine of Imaginary Number

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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 $-i$

$\blacksquare$


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