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 $-i$

$\blacksquare$

This article, or a section of it, needs explaining.
In particular: Investigate the multifunctional nature of Definition:Complex Inverse Sine and similarly for Hyperbolic Sine
You can help $\mathsf{Pr} \infty \mathsf{fWiki}$ by explaining it.
To discuss this page in more detail, feel free to use the talk page.
When this work has been completed, you may remove this instance of {{Explain}} from the code.

Sources

1968: Murray R. Spiegel: Mathematical Handbook of Formulas and Tables ... (previous) ... (next): $\S 8$: Hyperbolic Functions: $8.92$: Relationship between Inverse Hyperbolic and Inverse Trigonometric Functions