I know we have $Arc$ notation but my teacher told me $arc$ and $Arc$ having different . $Arcsin (x)$ means the first angle in $[-\pi/2 , \pi/2]$ which sine of it is $x$ and $arcsin(x)$ means all of the angles in $[-\pi/2 , \pi/2]$ which sine of it is $x$ .
I searched in the Google but didn't get this. I want to know that is it true ?
If you compute inverse $\sin$ and restrict solutions to that range then there will only be 1 angle anyway.
Capitalising the first letter conventionally indicates the principal value of an otherwise multi-valued function. It certainly occurs for complex valued functions such as Arg for the principal argument and Log for the principal log of $z$. It may not be a universal convention.