# What is the standard notation for $\arcsin$

I found a lecture notes that claims the following. Is this standard?

The notation $\overline{\text{arc}}\text{ sin }x$ is the inverse function of $\sin x$ restricted to $\left [ -\frac{\pi}{2},\frac{\pi}{2}\right ]$ and $\text{arc sin }x$ mean all those $y$ satisfying $\sin x=y.$

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$\arctan$, $\tan^{-1}$ are usually used. – Pedro Tamaroff Oct 26 '13 at 15:30
Why are you using \text{arc sin} rather than \arcsin? ($\arcsin$)? – Asaf Karagila Oct 26 '13 at 15:31
If I saw correctly, the notes has a space between the letters c and s. – forumreader Oct 26 '13 at 15:33

The most common notation used is either $\,\arcsin x\,$ or $\,\sin^{-1}x$.

When the desired value of $\,f(x) = \arcsin x\,$ is restricted to those values lying in $[-\pi/2, \pi/2]$, this is usually stated explicitly. I presume the lecturer introduced $\overline{\text{arc}}\sin x$ to spare the need from restricting the range of solutions repeatedly.

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So is $\arcsin x$ the set of all solutions to $\sin y=x$ or is it just some real number satisfying the equation? – forumreader Oct 26 '13 at 15:40
@amWhy I disagree. $\arcsin x$ is a function in its own right, defined over $[-1,1]$ and taking values on $[-\pi/2,\pi/2]$. – Pedro Tamaroff Oct 26 '13 at 15:47
I don't think I've said anything to contradict that, @Pedro. – amWhy Oct 26 '13 at 15:52
Your last comment. – Pedro Tamaroff Oct 26 '13 at 15:52
forumreader: Yes, $y= \arcsin x$ is the set of solutions to $\sin y = x$, where $y \in [-\pi/2, \pi/2]$ – amWhy Oct 26 '13 at 16:01