for 30 60 90 triangles, the ratio of the side lengths is [tex] \frac{1}{2} : \frac{ \sqrt{3} }{2} :1[/tex] 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 [tex]\frac{ \sqrt{3} }{2} :1[/tex] or [tex] \frac{ \frac{ \sqrt{3} }{2} }{1} [/tex] which equals [tex] \frac{ \sqrt{3} }{2} [/tex]
