You could use reach, STA, HTA, BB drop and front center to take a stab at rake. And you can derive front center if you have the rest of the numbers. If all the numbers were accurate your rake would be within 2mm.

In practice, some posted geometry is not accurate enough, but most modern bikes are designed on CAAD so they are more likely to be on. I tried figuring out some missing numbers on an old Cannondale and found that the published dimensions didn't work together.

But deriving trail without rake isn't going to work because all trail is is a "ratio" between rake and HTA.