The special right triangle lies in Quadrant 1 of the unit circle. A 30-60-90 right triangle contains a '2n' as the hypotenuse (r), 'n' as it's opposite side (y), and n radical 3 as it's adjacent side (x). Remembering that the radius for the unit circle is always one, we have to make 'r' equalling to 1. Divide all the sides by 2 in order to make the radius one. You will end up with the lengths: 1 (hypotenuse), 1/2 (opposite side), and radical 3 over 2 (adjacent side).
When dealing with a 45-45-90 right triangle, the radius must be 1. The hypotenuse of a 45-45-90 right triangle contains n radical 2, adjacent side being n, and opposite side also being n. In order to make the radius one, divide all sides by radical 2. The hypotenuse will be 1, the adjacent side will be radical 2 over 2, and the opposite side will also be radical 2 over 2.
It matches up with the unit circle. Using the 45-45-90 right triangle, we are able to see the ordered pair matching up with the 45 degree angle -> (radical 2 over 2, radical 2 over 2). Using the 30-60-90 right triangle, we are able to see the ordered pair matching up with the 60 degree angle -> (1/2, radical 3 over 2) and the 30 degree angle -> (radical 3 over 2, 1/2).
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