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Both refer to the inverse sine function.
The inverse sine (or arcsine) of x:
[tex]\mathsf{f(x)=arcsin(x)=sin^{-1}(x)}[/tex]
where x is a real number in the domain of the function:
[tex]\mathsf{-1\le x\le 1}[/tex]
and the arcsine function returns an angle in the interval [tex]\mathsf{\left[-\,\frac{\pi}{2},\,\frac{\pi}{2}\right]:}[/tex]
[tex]\mathsf{-\,\dfrac{\pi}{2}\le arcsin(x)\le \dfrac{\pi}{2}}\quad\longleftarrow\quad\textsf{range of the inverse sine function.}[/tex]
So if you see anywhere one of these expressions below
[tex]\mathsf{\theta=arcsin(x)~~or~~\theta=sin^{-1}(x)}[/tex]
then you should look for an angle [tex]\theta[/tex] that satisfies the following conditions:
[tex]\footnotesize\begin{array}{l}\bullet\end{array}\normalsize\begin{array}{l}\mathsf{sin\,\theta=x;} \end{array}\\\\\\
\footnotesize\begin{array}{l}\bullet\end{array}\normalsize\begin{array}{l}\mathsf{\dfrac{\pi}{2} \le \theta\le\dfrac{\pi}{2}.} \end{array}[/tex]
This angle [tex]\theta[/tex] is called the inverse sine of the real number x.
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Pay attention and do not mistake the arcsine function for the reciprocal of sine (which is cosecant); especially if you prefer or see that notation with an superscript -1. This one can be easily mistaken for an exponent:
[tex]\mathsf{sin^{-1}(x)=arcsin(x)\qquad\quad\checkmark}[/tex]
but the reciprocal is something like
[tex]\mathsf{\big[sin(x)\big]^{-1}=\dfrac{1}{sin\,(x)}=csc(x)\qquad\quad(!!)}[/tex]
and this last one has a total different meaning.
I hope this helps. =)
