Since the unit circle has a radius of 1 we know that r = 1 and that our angle is 60 degrees. Using sin and cos functions we can then find the x and y coordinates of point A. since 
 and
 and 
 (where
 (where 
 = the angle)
 = the angle) 
 because of the identity that cos(a) = x and sin(a) = y
 because of the identity that cos(a) = x and sin(a) = y
Using the angle and the previously described identity we can find the coordinates with cos(60) = x = 1/2 and sin(60) = y = 
 to get A =(x,y) =
 to get A =(x,y) = 
 .
.
From here we can work backwards and find that BC = x = 
 and that AC = y =
 and that AC = y =
 and finally we know that AB = r = 1 because it is a radius of the unit circle and must be 1.
 and finally we know that AB = r = 1 because it is a radius of the unit circle and must be 1. 
-This probably isn't the way you were taught to solve this problem, but it is how we do it in physics and real-life.