Question

What is the difference between arcsin and sin^-1 ?

Answer 1

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Both refer to the inverse sine function.

The inverse sine (or arcsine) of x:

$\mathsf{f(x)=arcsin(x)=sin^{-1}(x)}$


where x is a real number in the domain of the function:

$\mathsf{-1\le x\le 1}$


and the arcsine function returns an angle in the interval  $\mathsf{\left[-\,\frac{\pi}{2},\,\frac{\pi}{2}\right]:}$

$\mathsf{-\,\dfrac{\pi}{2}\le arcsin(x)\le \dfrac{\pi}{2}}\quad\longleftarrow\quad\textsf{range of the inverse sine function.}$


So if you see anywhere one of these expressions below

$\mathsf{\theta=arcsin(x)~~or~~\theta=sin^{-1}(x)}$


then you should look for an angle $\theta$ that satisfies the following conditions:

$\footnotesize\begin{array}{l}\bullet\end{array}\normalsize\begin{array}{l}\mathsf{sin\,\theta=x;} \end{array}\\\\\\ \footnotesize\begin{array}{l}\bullet\end{array}\normalsize\begin{array}{l}\mathsf{\dfrac{\pi}{2} \le \theta\le\dfrac{\pi}{2}.} \end{array}$


This angle $\theta$ 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:

$\mathsf{sin^{-1}(x)=arcsin(x)\qquad\quad\checkmark}$


but the reciprocal is something like

$\mathsf{\big[sin(x)\big]^{-1}=\dfrac{1}{sin\,(x)}=csc(x)\qquad\quad(!!)}$

and this last one has a total different meaning.


I hope this helps. =)