for 30 60 90 triangles, the ratio of the side lengths is
\(\frac{1}{2} : \frac{ \sqrt{3} }{2} :1\)
where the first is the shorter leg, the second is the longer leg, and the third is the hypotenuse (the longest side that doesn't form a 90 degree angle)
so the ratio of the longer leg to the hypotenuse is \(\frac{ \sqrt{3} }{2} :1\)
or
\(\frac{ \frac{ \sqrt{3} }{2} }{1}\)
which equals \(\frac{ \sqrt{3} }{2}\)
