If there were a trap there, it would have the same type of threaded fitting that your existing drain assembly has.
- cut the tailpiece a bit shorter
- install a 1 1/2" slip-joint nut and 1 1/2" x 1 1/4" reducing slip-joint washer onto the tailpiece
- insert the assembly all the way up through the hole in the sink
- screw the upper drain assembly flange, packed with plumber's putty, onto the top of the drain assembly
- lower the assembly such that the tailpiece enters the top of the drain pipe (it looks like it will line up)
- tighten the lower lock-nut, such that it pushes the tapered gasket up against and centered on the hole in the sink, from below. This will tighten the drain assembly to the sink, effecting a seal at the top with the plumber's putty under the upper drain assembly flange. Wipe off excess putty that squeezes out.
- slide the slip-joint washer and nut down to the threaded drain pipe fitting and tighten firmly (but carefully) with a wrench. P.S. hand tight is not good enough

picture from Google Images
p.s.s.- I am dubious of your presumption that a trap exists someplace downstream