Inverse Cosine of Imaginary Number

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Theorem

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


Proof

\(\displaystyle y\) \(=\) \(\displaystyle \cos^{-1} x\)
\(\displaystyle \leadsto \ \ \) \(\displaystyle \cos y\) \(=\) \(\displaystyle x\) Definition of Inverse Cosine
\(\displaystyle \leadsto \ \ \) \(\displaystyle \map \cos {\pm \, y}\) \(=\) \(\displaystyle x\) Cosine Function is Even
\(\displaystyle \leadsto \ \ \) \(\displaystyle \map \cosh {\pm \, i y}\) \(=\) \(\displaystyle x\) Cosine in terms of Hyperbolic Cosine
\(\displaystyle \leadsto \ \ \) \(\displaystyle \pm \, i y\) \(=\) \(\displaystyle \cosh^{-1} x\) Definition of Inverse Hyperbolic Cosine
\(\displaystyle \leadsto \ \ \) \(\displaystyle y\) \(=\) \(\displaystyle \pm \, i \cosh^{-1} x\) multiplying both sides by $\pm \, i$

$\blacksquare$


Sources