From my understanding, a paramedic is referred to the medical professional who stands by on the tennis court (or football field) and provide immediate care when someone is injured.

A trainer functions more like a coach. We usually get a trainer to help us exercise and avoid injuries when we work out in the gym.

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Note added at 3 hrs (2008-01-08 04:12:17 GMT)
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In addition to the coach, any athelete can also has his own trainer (who helps him practice or exercise during the regular training). For any sports team, there is also a team doctor. If an athlete looks like (or is) injured, the coach, trainer, and team doctor will run to the court first and check on him. If there is a serious injury, the paramedic will go up there.

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Note added at 22 hrs (2008-01-08 23:19:23 GMT)
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The answer came to my mind mostly because the asker mentioned "medic" in his question. In addition, I seem to remember seeing paramedics being called on the field when I was watching NFL on TV. Maybe the football players were way too violent so that only they need paramedics. Who knows? One final note, paramedics also ride on the ambulance.