The Goggled Helmet has a low chance (~5%?) to drop in a Arc chest. So it's a matter of Random Drop. You might have to run the Arc level a few times to get it, or the other Arc items.
Note: You're better served by clearing the main floor, hoping down to the Reactor, than back up again to re-set the Arc's enemies and chests. Don't try the save and re-load farming method. It doesn't work so well.