algebra precalculus - Pre-calculus method for finding the 'neat' value $sin(frac{pi}{3})=frac{sqrt3}{2}$?

If you have a unit circle and the Pythagorean theorem, how do you discover that $\sin(\frac{\pi}{3})=\frac{\sqrt3}{2}$? Finding the $1, 1, \sqrt2$ triangle seems more obvious. Do you consult a chart of previously-found Pythagorean triples and scale them to a unit hypotenuse? Do you have some reason (and, if so, what?) for wanting to know the sine whose cosine $=\frac{1}{2}$ and get lucky with a neat (as long as you don't mind surds) value? Do you use lengthy trial and error (historically, over centuries)? Or is there some other pre-calculus method than Pythagoras?