How can you expand arcsine, arccosine, and arctangent to accept complex/transcendental numbers as inputs? (ex. $\sin(x) = \pi$)
I reorganized Euler's formula $e^{ix} = \cos(x) + i \sin(x)$ into the form:
$\sin(x) = \frac{e^{ix} -e^{-ix}}{2i}$,
which is the standard expansion of sine into the complex numbers, and then I tried to solve for $\sin(x) = \pi$:
$e^{ix}-e^{-ix}=2\pi x$.
Multiply both sides by $e^{ix}$ and moves all parts to one side:
$(e^{ix})^2 - 2\pi ie^{ix} - 1 = 0$
Then, solve for $e^{ix}$ and $x$:
$e^{ix} = i(\pi \pm \sqrt{\pi^2 - 1})$ and $x = -i \ln|i(\pi \pm \sqrt{\pi^2 - 1})| $
But, how can you simplify this into a generalized form for arcsine, arccosine, and arctangent?