# Inverse Cosine of Imaginary Number

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## Theorem

- $\cos^{-1} x = \pm \, i \cosh^{-1} x$

## Proof

\(\displaystyle y\) | \(=\) | \(\displaystyle \cos^{-1} x\) | |||||||||||

\(\displaystyle \implies \ \ \) | \(\displaystyle \cos y\) | \(=\) | \(\displaystyle x\) | Definition of Inverse Cosine | |||||||||

\(\displaystyle \implies \ \ \) | \(\displaystyle \cosh \paren {i y}\) | \(=\) | \(\displaystyle x\) | Cosine in terms of Hyperbolic Cosine | |||||||||

\(\displaystyle \implies \ \ \) | \(\displaystyle i y\) | \(=\) | \(\displaystyle \cosh^{-1} x\) | Definition of Inverse Hyperbolic Cosine | |||||||||

\(\displaystyle \implies \ \ \) | \(\displaystyle y\) | \(=\) | \(\displaystyle - i \cosh^{-1} x\) | multiplying both sides by $-i$ |

$\blacksquare$

## Sources

- 1968: Murray R. Spiegel:
*Mathematical Handbook of Formulas and Tables*... (previous) ... (next): $\S 8$: Hyperbolic Functions: $8.94$: Relationship between Inverse Hyperbolic and Inverse Trigonometric Functions