When to use what inverse trig?

Question

When do I use $\arcsin$ and when do I need to include all of the outcomes? My gut feeling is if you have an equation like $\sin(x)=0$, then $x=0,\pi,2\pi...$ whilst if you are using it in integration and any other situation you use $\arcsin$. Is this correct if either way why?? thanks

Answer 1

Usually, arcsin(x) is defined as the number y in the range $[-\frac{\pi}{2},\frac{\pi}{2}]$ with the property $sin(y) = x$. With this restriction, the inverse of sin(x) is well-defined. Solving the equation sin(x) = 0 is something else, here you do not restrict the range of the possible values, and you get infinite many solutions.

If arcsin(x) occurs in an integral, the definition I described should be meant.