This http://2e5.com/plotter/V/design/ has got a great discussion about the maths behind these kinds of plotters. The most basic maths is simple pythagoras  treating the hanging V as two rightangle triangles.
To calculate the cartesian X position you need to know the width of the machine (called pageWidth in the code) and the lengths of the two strings (I call these aPos and bPos in the code).
https://github.com/euphy/polargraph_server_a1/blob/master/util.ino#L225
To calculate the cartesian Y position you need to know the cartesian X position (called cX in the code) and the length of either one of the two strings (I use a lefthand string in the code, and call it aPos).
https://github.com/euphy/polargraph_server_a1/blob/master/util.ino#L225
The code describes explicitly how this works and it'll make sense if you remember that pow(x, 2) is simply "x squared"). I'll try to put it into words.
Get X:
cartesianX = (pageWidth squared  bPos squared + aPos squared) / (pageWidth*2)
Get Y:
cartesianY = square root of (aPos squared  cartesianX squared)
sn
