This uses trigonometry because of the Law of Sines (I've never heard it called the "Sine Rule" before, it took me a moment to figure out what you meant).

The proof of the Law of Sines doesn't actually use any other trigonometry, so it strikes me that if this is actually the proof in question, you could get rid of the trig part by, essentially, embedding the proof of the Law of Signs into the proof.