Finding Inverse Trig Functions

I know this may sound elementary...but trig has always been a frustration of mine. What are some good tricks to know or techniques on figuring these out? I mean I know all the common angles in the first quadrant, but I'm not well versed after that. I think therein lies a problem. Is it just a matter of memorizing the unit circle, restricted domains of the inverses, and the following identities?

For example. I know this is wrong but this what I tend to do.

Until I plug into the above formula, do I discover I'm wrong.

Is there any way to avoid these fallacies? or is it to just memorize the above?