sin(y) - y = x√(2)

[wheepep]

sin(y) - y = x√(2)
solve for y

 

[skeeter]

What makes you think a closed-form solution for y can be found?

 

[Country Boy]

Replacing \(\displaystyle sin(y)\) by \(\displaystyle \frac{e^{iy}- e^{-iy}}{2i}\) you might be able to get a solution in terms of the "Lambert W function" https://en.wikipedia.org/wiki/Lambert_W_function.