Do not use 0,0 for any local point. That will send you into all four quadrants with positive and negative coordinates.
There are a number of ways to relate the coordinates but why bother, find your UTM number and stick with them, move your figure calculated in autocad to the UTM position. The command is M. All you need to do after that is figure out the rotation from your calculated figure to UTM grid. Depending were you are in the zone it can be a minimal rotation or a multiple degree rotation. That can be easily figured out in Trimble for an astronomic to grid rotation, but when you locate actual monuments it will probably be slightly different. Keep your figure as a block, you can even copy, it move it, block it and wait until your rotation information is determined before exploding it.
As far as new coordinates from a position it’s a simple formula:
If you have a calculator for the above example it’s
easting=605605.481-(sin of bearing x 3909.08)=601730.683
northing=5344222.851+(cos of bearing x 3909.08)=5344739.420
Checkbook math. If you want to dig deep into UTM you will need to geodetically calculate latitudes and longitudes forward and then convert to UTM, but that’s not necessary for small projects like this one.
You can set your UTM zone in Autocad and then you will have lats, longs, Northings, Eastings, all available in Autocad and Trimble. Makes for easy checks between the databases.