How to say "Sorry I'm a little sick today" in korean or "How sick are you?"
I know how to "I'm sick" in Korean but some people worry about that. What are some phrases of that response like "When did you get sick?" or "Is it that bad?" or "Have you taken your medicine?"

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If you are saying to your friend or younger people, then you can say like this

When did you get sick? => "언제부터 아팠어?" 
"Is it that bad?" => "많이 안좋아?" or "많이 아파?"
"Have you taken your medicine?" => "약은 먹었어?"

If you are saying to elder people, then you can say like this

When did you get sick? => "언제부터 아프셨어요?" of "언제부터 편찮으셨어요?"("편찮으셨어요" is very honorific expression of "아프셨어요")
"Is it that bad?" => "많이 안좋으세요?" or "많이 편찮으세요?"
"Have you taken your medicine?" => "약은 드셨어요?"

Answer

Additional : Are you feeling better?  좀 나아졌어? 어때?