When studying complex analysis, we realize that trigonometric functions are nothing but exponentials, and we can define real trigonometric functions in terms of complex exponentials. I was wondering if we can apply this logic to define inverse trigonometric functions (arcsin, for example) in terms of complex logarithms, who are the inverse functions of complex exponentials. Can we? Is it appropriate?
Yes absolutely! See here, where the logarithmic forms of $\arcsin$ etc are given. There may be issues defining the domain, since the complex log function has a fair amount of subtlety in its definition. In a related but slightly simpler vein, the inverses of the hyperbolic trig functions can also be written in terms of $\log$, but without $i$ showing up. If you're familiar with hyperbolic trig, it might be a useful exercise to try and derive these inverses on your own, then see how you could apply your arguments to $\arcsin$.