I would just do away with the sin^{-1} notation (as, it seems, many textbooks already do) since we have the perfectly acceptable alternative "arcsin".
It's also not a very good notation since it's trying to imply that the sin function has an inverse, but it doesn't. That's why "sin^{-1}(sin(x)) = x" is not even right in general. The inverse only exists on specific subintervals, and it's also off by a multiple of pi, depending on that subinterval. "arcsin" is then defined as the inverse of sin, restricted to the interval [-pi/2,pi/2].
Of course, the bigger issue here is that f^n for any function f is inherently ambiguous, because it could refer either to the (pointwise) multiplication operation or to the composition operation.
This is a very good point. The sine function obviously doesn't have a technical inverse on any interval where it has two values. The notation does make me forget this sometimes.
Of course, the situation is different from many other functions without inverses, because the set of all valid inversions can be trivially generated from one solution. Just put a mirror at pi/2 and -pi/2.
I would just do away with the sin^{-1} notation (as, it seems, many textbooks already do) since we have the perfectly acceptable alternative "arcsin".
It's also not a very good notation since it's trying to imply that the sin function has an inverse, but it doesn't. That's why "sin^{-1}(sin(x)) = x" is not even right in general. The inverse only exists on specific subintervals, and it's also off by a multiple of pi, depending on that subinterval. "arcsin" is then defined as the inverse of sin, restricted to the interval [-pi/2,pi/2].
Of course, the bigger issue here is that f^n for any function f is inherently ambiguous, because it could refer either to the (pointwise) multiplication operation or to the composition operation.