Standalone 咳 is rare in speech - the everyday word for cough is the compound 咳嗽. Solo 咳 mostly appears in medical contexts (止咳 'suppress cough', 咳血 'cough up blood') and in descriptive prose.
mistakes
Don't confuse the two readings: 咳 (rising tone) = to cough; 咳 (level tone) = the sigh interjection 'alas / oh well'. Same character, different jobs.
Left radical 口 marks 咳 as a mouth-action: coughing forces air out from the throat past the lips. The mouth radical groups 咳 with other vocal-tract verbs like 喊 (shout), 吼 (roar), 吐 (spit) - all sharing this three-stroke square on the left.
Right phonetic 亥 supplies the sound - hài shifted to ké / hāi via tone and onset drift. The same 亥 phonetic series surfaces in 孩 (child), 该 (ought), 骇 (alarm) - recognizing 亥 unlocks all four readings at once.
