Someone who's insincere is a little bit fake or phony. Your classmate's insincere apology for interrupting you probably sounds like she doesn't really mean it.
When you're insincere, you're not being completely honest. A losing politician often gives an insincere-sounding concession speech, and a teacher who's having a terrible day might wear an insincere smile on her face. The adjective sincere, "coming from genuine, honest feelings," is at the root of insincere — the Latin prefix in, or "not," is combined with sincerus, "sound, pure, or whole."
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