$\sin x=\sin y$ mis-understanding $\arcsin(x)$ is the inverse function of $\sin(x)$, this means that:
$$\arcsin(\sin(x))=x$$
If I want to solve the equation:
$$\sin(x)=\sin(y)$$
using this link, $x=\pi - y$ for instance.
but, also we can say that:
$$x=\arcsin(\sin x)=\arcsin(\sin y)=y$$
which is for me a contradiction. Where is the missing point in my writings?
 A: WHat's wrong is your very first sentence. The function $\sin:\Bbb{R}\to\Bbb{R}$ DOES NOT have an inverse. It is a periodic function, so not injective (one-to-one) so definitely cannot be inverted.
On the other hand, if you restrict the domain and target, you get $\sin|_{[-\pi/2,\pi/2]}:[-\pi/2,\pi/2]\to [-1,1]$. This restricted function (which is different from the usual $\sin$, since we changed the domain and target) is invertible, and it's inverse is called $\arcsin:[-1,1]\to [-\pi/2,\pi/2]$. So,
\begin{align}
\arcsin:= \left(\sin|_{[-\pi/2,\pi/2]}\right)^{-1}.
\end{align}

Some remarks: we can restrict the $\sin$ function to other intervals and still get invertible functions. FOr example, $\sin|_{[5\pi/2,7\pi/2]}:[5\pi/2,7\pi/2]\to[-1,1]$ is also invertible, but its inverse function is NOT $\arcsin$, it is $\arcsin +2\pi$. There are actually infinitely many other intervals we can restrict to and get an invertible function, and it turns out the inverses differ by multiples of $\pi$
The name $\arcsin$, rather than $\sin^{-1}$ is thus appropriate. The function $\sin$ is not invertible, and $\arcsin$ refers to the inverse of a very specific restriction of $\sin$. Also, it is purely convention that we define $\arcsin$ by restricting $\sin$ to $[-\pi/2,\pi/2]$, the so-called principal branch.
