How to find how many degrees player is looking away from centre of object? / Where player is looking at on a plane

The above linked Imgur album should show what I mean by this. I want to be able to tell which part of the grey square the player is looking at. I assume I would do this by saying the actor is looking X amount left or right of the centre, and Y amount up or down from the centre.

How would I achieve this?

Im not sure if its exactly what you need but there is a node Find Look at Rotation, it takes two vectors in this case it would be your players position and a the end point of trace line to where you need it to connect. This should give you the rotation you require but Im not sure if it can return it in the method you mention in your images since its a rotation.