Hello 

Thanks a lot for trusting your question to me. 

Strep throat normally doesn't present with cough, actually if cough is referred as a symptom at initial presentation, other diagnosis must be consider.  

If your cough was not part of your initial presentation, it can be explained by a couple of reasons. 

Strep throat can cause complications, such as a condition called parapharingeal abscess (collection of pus with in your throat). This complication presents as persistent fever, persistent pain. If the abscess spontaneously ruptures, there
Is sudden improvement in symptoms, but the pus collection can be accidentally swollen an cause cough . 

Z pack is an antibiotic called azithromycine, some step throats have been found to have resistance to this medication. I don't think this is your case since you are improving. 

You should be fine, your symptoms are improving and cough could mean a secondary viral infection (nothing to do about that) and a rupture of a parapharingeal abscess (which means it has resolved). 

If symptoms are persistent or worsen and if other symptoms start, such as fever, breathless sensation or rash. I recommend you get a new medical evaluation. 

Hope I've been of some help.