I think of it like this: I'm requesting a sick day. Period. I'm not telling you if it is me or a family member or my cat. I just need time - paid time - to take care of an illness. I know some places have policies about taking more than so many consecutive days (3 days or more may require a doctor's note) - but, otherwise, it is my sick time and I'll take it for my cat or just because I need some time to reflect or whatever.