There are two types of coughing. One is where the head and neck are outstretched so they're parrallel to the floor, the cat is crouched (not sitting) and does individual coughs in a regular candance, and sometimes the head will swing side to side slowly, but the chin is as far forward as possible. This allows the lungs to 'huff' something out of the throat.
Then there's the hairball, puke, "I ate grass" cough with the head tipped down and the nose pointing to the floor, usually with the back arched, and done from a sitting position. The sides heave, the cough is deep and the eyes are usually closed. This is when something has to come up out of the stomach, or has to clear from way down in the lungs.
The first isn't a concern, I call it dry hacking (I've only seen one hairball in 18 years that came up with a dry hack, usually nothing comes up, the cat coughs till it suddenly swallows and then gets up and walks away).
The second is when the cat has to get something out of it's body.