In the Principal Translations, the 4th definition of using the word "since" like the word "because" has an error in the example sentence (as far as I know). The sentence " I'll cook dinner, since you're so busy." should not use a comma. The first part/clause of the sentence "I'll cook dinner" does not have a negative verb, so you don't use a comma. I think the correction should be "I'll cook dinner since you're so busy." Please tell me if I'm wrong, thanks!