Anyone vaguely familiar with saurischian dinosaurs or pterosaurs (or birds for that matter) will know about pneumatic bones. Extensions of the lungs penetrate the bones and fill part, or a great deal, of their internal volume leaving them with hollow vertebrae, humeri, and even ribs and furculae. However pretty much all tetrapods (I’m sure there’s one exception somewhere I don’t know about or I’d say all) have a pneumatic skull. The sinus are a large cavity inside a bone (OK series of bones in the case of the skull) that attach to the lungs (if from the other end of the trachea). These can be quite complex however, and this is not as much as a cheat as it may sound.
The photo above if from the palate of a mammal. Although there’s no scale in the image I think it is reasonably clear that in places the bone is really rather thin (especially the broken piece at the top centre) and around the middle there are obviously fine filigrees of bone and large spaces beyond. It may be a major surprise to learn that this is the skull of an elephant, and yes those bone really are just a mm think or less – it’s quite a delicate structure and with a lot of space behind that first layer of bone. Despite the size of the animal and the thickness of the bone in other places in the skull, the sinus is still a massive cavity with some very fine bone in there that is, basically, air filled.