So I can't find anything on it in the faq or rules reference, does tactician apply to a ships auxiliary firing arc?

yes.

Yes, Tactician works. Primary weapons work in auxiliary arcs.

Yes, Tactician works. Primary weapons work in auxiliary arcs.

Tactician doesn't care what type of weapon you're firing, just that the enemy is in arc. Auxiliary arcs are arcs, to Tactician works there (at Range 2 of course). If you have, say, a Heavy Laser Cannon and shoot it at someone in arc, Tactician still works.

Auxiliary arcs are firing arcs for all things that care about firing arcs. Outmaneuver, tactician, and all the rest.