What is the difference between $\operatorname{Arcsin}$, $\operatorname{arcsin}$, $\operatorname{Sin}^{-1}$, and $\sin^{-1}$? This is a homework problem I have. I don't remember learning the difference, and searching hasn't helped explain the difference between the capitalization.
While I'm almost certain the other answers are correct about $\arcsin$ being equivalent to $\sin^{-1}$, the majority of sources I can find contradict the answers given by MarianD and Polygon:


The Math Forum's "Ask Dr. Math": Inverses of Trigonometric Functions
MathWords.com: Inverse Trigonometry
Dummies.com: How to distinguish between trigonometry functions and relations
MathWorld: Inverse Trigonometric Functions
SparkNotes: Inverse Trigonometric Functions


According to these sources
$$\arcsin x =\sin^{-1}(x) = \{ \theta \in\Bbb{R} : \sin \theta = x\} ,$$
whereas 
$$\operatorname{Arcsin} x =\operatorname{Sin}^{-1}(x)= \theta,$$
where $\theta$ is the unique angle between $-\pi/2$ and $\pi/2$ (inclusive) such that $\sin\theta = x$, which is often called the principal value.

Side note

I'd like to comment that in practice, in most cases the lower case variants are used to denote the principal value, and the upper case variants are not used at all. However, when they are used, this does appear to be the correct answer regarding their meaning.
$\operatorname{Arcsin}$ is used when showing $\sin^{-1}x$ as a relation whereas $\arcsin$ is used when showing $\sin^{-1}x$ as a function. If you put the function sin^-1(x) into Desmos or a graphing calculator set to radians, you can see the graph of the arcsine function (inverse function of $\sin x$). You will also see that the line stops abruptly at $x=-1$ and $x=1$ and $y=-\pi/2$ and $y=\pi/2$. Since there cannot be two $y$ values for any one $x$ value, the graph stops. We denote this segment as the &quot;arcsine&quot;. But if the graph continued as a relation, it would look like a sine function stretching from $y=-\infty$ to $y=\infty$. We denote this as &quot;Arcsine&quot;.
I hope someone finds this useful.