In the 1977-78 school year, a student of mine named Ed Williams showed me his method for remembering the coordinates of the points in the first quadrant on the unit circle.
His method yields the coordinates for the angles (in degrees): 0, 30, 45, 60, and 90.
You can use symmetry to determine the values of the other points on the unit circle.
Ed made the denominators of all the points on the unit circle equal to 2.
Beginning on the positive x-axis at 0 degrees, and continuing to 90 degrees,
the numerators of the y-coordinates are:
, 
,
, 
,
and 
.
Then the numerators of the x-coordinates beginning with 90 degrees and going back
down to 0 degrees are:
, ,
, ,
and .
Simplify the square roots and you are done (see the answers below).
