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.

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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.

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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?”

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Applying the Pythagorean theorem, we get this:

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Clearly there is only room for one quantity here: if “something” plus 1/4 is 1, then that something had better be 3/4!

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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!