Domain/Range of Inverse Trig Function

This question is difficult to put into words, but here we go...

1) If the domain of y = sin(x) is all possible x values...

2) And if the range of y = sin(x) is all possible y values...

...When looking at the inverse of this function...

3) Wouldn't the domain of x = sin(y) be all possible **y** values?...

4) And, similarly, the range of x = sin(y) be all possible **x **values?...

...Regardless of the notation being y = arcsin(x) or y = sin^-1(x)

I ask because I'm confused as to why the y values of x = sin(y) are always referred to as the range in this situation (and x values as the domain). Maybe I'm missing something huge, but it seems to me that the common notation (y = arcsin(x)) should not take precedence over the actual *meaning* (x = sin(y)) when deciding upon the domain and range of a function. Wow, that was difficult to express! (Worried)