Here’s my take on how to “see” the sides of a 30-60-90 triangle, without “memorizing” anything. Students learning trigonometry are expected to know the side ratios for “special angles,” and so it’s useful to know how to work them out quickly.
First, students should learn to recognize that a 30-60-90 triangle is half of an equilateral triangle. Thus, if the hypotenuse is 1 (thinking of a unit circle here), then the shorter leg must be 1/2.
Now let’s deal with that pesky longer leg. Assuming we’ve worked out this length using the Pythagorean theorem before, it won’t come as a surprise that it’s “the square root of something.” But what is that “something?”
Applying the Pythagorean theorem, we get this:
Clearly there is only room for one quantity here: if “something” plus 1/4 is 1, then that something had better be 3/4!
Thus the long leg is the square root of 3/4. The advantage of writing the leg in this way is that we can do a quick mental check: the square root of 3/4 squared, plus 1/2 squared, is indeed equal to 1 squared, as 3/4 + 1/4 = 1. Done!