Hi,thank you for posting your query on health care magic.
In case of
sinus infection,secretions may pass from the nose to the throat(called Posterior nasal Drip) and irritate the throat resulting in dry cough.
You may find relief by taking Anti-histaminic(like Cetrizine),Oral Decongestant if you are not Hypertensive(like containing
Phenylephrine),Nasal Decongestant Drops (like containing
Oxymetazoline),Cough Suppresant(like containing
Dextromethorphan),Analgesic with antacid(it there is headache) and Saline
nasal Spray along with the Antibiotic.
Also do steam inhalation.Do warm saline gargles.Drink plenty of fluids(warm).Avoid cold,oily and very sweet food as they increase mucus production.Hot tea with ginger,honey and lemon will also give you relief.Avoid dust,smoke and other pollutants.Avoid coughing very forcefully as it may tear tiny blood vessels due to increased pressure leading to bleeding.
Hope this answers your question.Get well soon.