Agreed "using trigonometry" is potentially misleading. After reading the proof, the only 2 senses in which "trignometry" is being used are:
1. The term "sin a" is used to denote the ratio opposite / hypotenuse. But this can be considered a purely notational convenience. They could have called it "foo a" and nothing would change, or they could have inlined the referred-to ratio everywhere.
2. The law of sines is required. But the proof of this law [1] also boils down to nothing more than the ratio definition and some algebra.
So afaict no circular logic is being used, but at the same time it doesn't seem to be doing anything previously thought to be impossible, unless there was a previous belief that the law of sines could not be used in a proof, which would be a strange belief to hold. I see it simply as a creative, unexpected proof.