Coarse is only an adjective. Its main senses in today’s English are (1) of low quality, (2) lacking refinement or vulgar, and (3) rough in texture or composed of large particles. For example, a movie regarded as obscene or lowbrow might be called coarse, as might a person who speaks in a rude or off-color way. In the third sense, the adjective’s applications are broad. Most commonly, it tends to describe rougher varieties of sand, asphalt, soil, and fabric.
Course, which works only as a noun or a verb, has many definitions. It refers to paths, durations, academic classes, golf playing fields, and parts of incrementally consumed meals, among other things. As a verb, it means, primarily, to move along a course. Blood, for instance, is often described as coursing through the veins. The word also appears in the idiom of course, which means naturally, obviously, or certainly.
Based on the study of the use of coarse language by Canadian fishermen, Menzie (1991) concluded that coarse language can reinforce a culture of gender inequality. [Handbook of Gender Research in Psychology, Joan C. Chrisler, Donald R. McCreary]
Double the size of fine or medium, and you’ve got coarse bulgur, which has an accordingly more substantial flavor and texture. [Bob’s Red Mill Cookbook, Miriam Backes]
Many of Man Ray’s pictures of women have become archetypes of a certain kind of photographer’s gaze ever since: voyeuristic without being too cruel, randy without being coarse. [Financial Times]
To maintain a straight course and good stroke mechanics, you must remain mentally alert hour after hour. [Open Water Swimming, Steven Munatones]
In the syllabus for an Applied Mathematics course, students are told to write their problem sets individually. [Boston Globe]
In the veins of turtles coursed a sweet lassi that had to be drunk as soon as it spurted from their necks, because it coagulated in less than a minute. [Life of Pi, Yann Martel]