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What activity were you engaged in when the symptoms started?
Have you tried taking any medicines for this?
I had asked for some further information and have not heard back.
The nature of the activity being done when the symptoms started may provide a clue regarding the cause of the pain.
The fact that the pain is made worse by coughing would suggest that the underlying problem is actually in the neck or upper chest and affecting the nerves that travel to the arm, rather than a problem within the arm, itself.
In a young adult, it would be more common for the underlying problem to be a strain/sprain or some minor inflammation. Although an examination would help to delineate the underlying cause, it would be fine to first try an over the counter anti-inflammatory medicine, such as ibuprofen or naproxen. It also would be fine to apply ice to the neck, upper chest, and shoulder for about 15-20 minutes, 2-3 times per day.
If I can provide any further information, please let me know.