Inverse Hyperbolic Cosecant of Imaginary Number
Jump to navigation
Jump to search
Theorem
- $\map {\csch^{-1} } {i x} = -i \csc^{-1} x$
Proof
\(\ds y\) | \(=\) | \(\ds \map {\csch^{-1} } {i x}\) | ||||||||||||
\(\ds \leadsto \ \ \) | \(\ds \csch y\) | \(=\) | \(\ds i x\) | Definition of Inverse Hyperbolic Cosecant | ||||||||||
\(\ds \leadsto \ \ \) | \(\ds i \csch y\) | \(=\) | \(\ds - x\) | $i^2 = -1$ | ||||||||||
\(\ds \leadsto \ \ \) | \(\ds \map \csc {i y}\) | \(=\) | \(\ds x\) | Hyperbolic Cosecant in terms of Cosecant | ||||||||||
\(\ds \leadsto \ \ \) | \(\ds i y\) | \(=\) | \(\ds \csc^{-1} x\) | Definition of Inverse Cosecant | ||||||||||
\(\ds \leadsto \ \ \) | \(\ds y\) | \(=\) | \(\ds -i \csc^{-1} x\) | multiplying both sides by $-i$ |
$\blacksquare$
Sources
- 1968: Murray R. Spiegel: Mathematical Handbook of Formulas and Tables ... (previous) ... (next): $\S 8$: Hyperbolic Functions: $8.103$: Relationship between Inverse Hyperbolic and Inverse Trigonometric Functions