Inverse Hyperbolic Cosine of Imaginary Number
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Theorem
- $\cosh^{-1} x = \pm \, i \cos^{-1} x$
Proof
\(\ds y\) | \(=\) | \(\ds \cosh^{-1} x\) | ||||||||||||
\(\ds \leadsto \ \ \) | \(\ds \cosh y\) | \(=\) | \(\ds x\) | Definition of Inverse Hyperbolic Cosine | ||||||||||
\(\ds \leadsto \ \ \) | \(\ds \map \cosh {\pm \, y}\) | \(=\) | \(\ds x\) | Hyperbolic Cosine Function is Even | ||||||||||
\(\ds \leadsto \ \ \) | \(\ds \map \cos {\pm \, i y}\) | \(=\) | \(\ds x\) | Hyperbolic Cosine in terms of Cosine | ||||||||||
\(\ds \leadsto \ \ \) | \(\ds \pm \, i y\) | \(=\) | \(\ds \cos^{-1} x\) | Definition of Inverse Cosine | ||||||||||
\(\ds \leadsto \ \ \) | \(\ds y\) | \(=\) | \(\ds \pm \, i \cos^{-1} x\) | multiplying both sides by $\pm \, i$ |
$\blacksquare$
Sources
- 1968: Murray R. Spiegel: Mathematical Handbook of Formulas and Tables ... (previous) ... (next): $\S 8$: Hyperbolic Functions: $8.95$: Relationship between Inverse Hyperbolic and Inverse Trigonometric Functions