Basement Rough-In

I am beginning a basement finishing project and would prefer to have a full bathroom consisting of a sink, toilet, and shower. However, I am having trouble determining the configuration of the rough-in.

There are two pipes (a 1-1/2" and a 2") and a closet flange. I am wondering if the 2" pipe is a vent for the toilet with the 1-1/2" pipe for a sink or the 2" for a sink and the 1-1/2" for a shower. Attached is a picture of the site.

What is the best way to determine the rough-in configuration?

Well, a shower requires a 2" drain line, so that is probably your shower/tub drain. The 1.5" is for the sink, and the flange is of course, the toilet. My unprofessional opinion. Note, the 2" one should have a trap, you should be able to see water down it, unless it has dried out. The 1.5" one, if for a sink, would expect a trap under the sink, not in the floor.

pipe

It is hard to say what the 1 1/2" pipe is for. It is too small for a shower, is in the wrong place for a tub, and is too far from anything else not to have a vent if it is a riser from a "P" trap.

Without knowing the intent of the original plumber, it's hard to say.

The 1.5" pipe could be a vent for a buried p-trap, or it may be a lav drain.

The 3" pipe could be a vent for the toilet, or it could be the waste from the second floor.

If it's the waste from the second floor, then the 2" may be the toilet vent.
In some cases, the 2" could be the toilet vent, and the waste for a lav, that is wet vented over the toilet vent.

Without seeing the rest of the piping, it is hard to tell.

The 3" pipe is waste from the floors above. So that must mean that the 2" pipe is a vent for the toilet and the 1 1/2" pipe is for a lav since the it is too small for a shower. Unfortunately it looks like there is no way of having a full bath without breaking up the concrete. Thank you.

pipes

"Standard" procedure would be that the 2" line is the vent for the toilet and the drain for the lavatory. Only investigation can tell if the 1 1/2" pipe is a trapped inlet, or a vent for a buried trap.