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.
 A: 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.
A: We may symbolically write:

*

*$\arcsin\ $ = $\sin^{-1}\, $

*Arc$\sin\ $ = Sin$^{-1}\, $
because

*

*$\arcsin\ $ and $\sin^{-1}\, $ are only two different notations for the same function,


*

*Arc$\sin\ $ and Sin$^{-1}\, $are only two different notations for the same maping.
Definition: The Arcsine of $x$, denoted $\operatorname{Arcsin}(x)$, is defined as "the set of all angles whose sine is $x$". It may be interpreted as a one-to-many relation.
Definition: The arcsine of x, denoted $\arcsin(x)$, is defined as “the (only) angle from the closed interval $[-\pi/2, +\pi/2]$ whose sine is $x$”. It may be interpreted as a one-to-one relation.

So, for example
$$\arcsin\left(\frac1 2\right)= \sin^{-1}\left(\frac1 2\right) = \frac{\pi}{6} $$
while
$$\operatorname{Arcsin}\left(\frac1 2\right)= \operatorname{Sin^{-1}}\left(\frac1 2\right) = \left.\left\{\frac{\pi}{6} + 2k\pi, \frac{5\pi}{6} + 2k\pi\ \right| \ k \in \mathbb Z\right\} $$
See also definitions and illustrations of these functions / mappings in The Algebra Help e-book on MathOnWeb.com site.

Edit:
It seems that definitions for capitalized and lowercase names are swapped. Several other sources — see the jgon's answer — defines them in the opposite manner.
A: Sin$^{-1}$ (with a capital S) returns every value you could put into the sine function to get your input. So $\operatorname{Sin}^{-1}(\sqrt2/2) = \{\cdots\frac{-7\pi}4,\frac{-5\pi}4,\frac\pi4,\frac{3\pi}4,\frac{9\pi}4,\frac{11\pi}4\cdots\}$.
But with a lowercase s, it only returns values between $\frac{-\pi}2$ and $\frac\pi2$.
So $\sin^{-1}(\frac{\sqrt{2}}2)$ is just $\frac\pi4$.
The notation with the capital S is rarely used because it is not a function; one input gives you infinitely many outputs.
A: $\operatorname{Arcsin}$ 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 "arcsine". 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 "Arcsine".
I hope someone finds this useful.
