I can't recommend you use wood glue. Wood glue will set up long before the sinew cures out. In effect, it will waste the benefit that the shrinkage of the sinew imparts to the bow as the moisture cures out. Yes, it will back the bow and prevent splinters from lifting. But if that's all you want, back it with rawhide, silk, light canvas, strips of brown paper bag, or even effing glass.
If you are looking to improve the tension side of the bow by using sinew, go all the way and use hide glue. Hide glue shrinks up just like the sinew does and the two work in tandem to "pre-stress" the tension (back) side of the bow. Most places selling sinews also sells hide glue, but avoid the hide glue in the bottle that you may see at your lumberyard or hardware store. It has been treated with chemicals to keep it liquid and prevent molding and spoilage. It's really no longer the same stuff as plain hide glue.
If you can't find hide glue, go to the grocery store and purchase a package of Knox unflavored gelatin (or the generic version if you are as cheap as me). It's colorless, odorless, and tasteless hide glue. The odorless factor is an overwhelming point in it's favor, especially if you have 1 or more female types that inhabit your home. Trust me on this one.