Coarse means rough in texture or crude in manner — coarse sand, coarse language. Course covers a route, a series of lessons, part of a meal, and the phrase "of course". Only one of them describes texture, and it is the one with the A.
The core difference
Homophones with one narrow meaning and one very broad one.
- coarse — rough, crude: coarse grain, coarse humour.
- course — a route, a programme of study, a meal part, a golf course, "of course".
The reliable check
Coarse has an A, and so does rough… which does not help. Better: coarse is almost always an adjective describing texture or manner. If the word is a noun, it is nearly always course.