algebra precalculus - Imaginary $cos^{-1}$ value significance?

When I was bored in AP Psych last year, I jokingly asked myself if there was a cosine inverse of $2$. Curious about it, I tried calculating it as follows:
$$
\begin{align*}
\cos (x) &= 2 \\

\sin (x) &= \sqrt{1 - \cos^2(x)} = \sqrt{1 - 4} = \pm i \sqrt{3}
\end{align*}
$$
Then, by Euler's formula, you have
$$
\begin{align*}
e^{ix} &= \cos (x) + i \sin (x) \\
e^{ix} &= 2 \pm\sqrt{3} \\
ix &= \ln (2 \pm \sqrt{3}) \\
x &= \boxed{-i \ln (2 \pm \sqrt{3})}

\end{align*}
$$

So, there was a way to calculate the inverse cosine of numbers whose magnitude is greater than $1$ (this was verified on Wolfram Alpha). To what extent is this kind of calculation valid? Does it have any interesting applications/implications in math, or any other subjects? Thanks. :)

Edit I just realized this is very easily explained by $2\cos (x) = e^{ix} + e^{-ix}$, but I'm still curious if this has any significance/intuition.