It depends on the nature and scope of the illness.
A Choleh she’ain bo sakanah is allowed to take such medicine Aino reuii leachila adam or She’lo ki’derech achila
Cough syrup is derech achila.
Therefore, what I would do if I got sick would be take each teaspoon of cough syrup and mix it with 2 oz. of water or juice. You are allowed to be mevatel a safek issur, which is what the cough syrup is.