In your original post you said "On unit circle, 1/2 is equal to pi/6 and 5pi/6". Now you know better than that! "1/2= 0.5" while pi/6 is about .52. What you meant was that sin(pi/6)= 1/2. LCKurtz was suggesting that you be more careful about saying what you mean!
On the unit circle, sin(t) is the y coordinate of a point at distance t from (1, 0) measured around the circle. Two points on the same horizontal line will have the say y value. If you draw a horizontal line at y= 1/3, it crosses the line twice, at points that are symmetric with respect to the x-axis. Yes one of the points is a distance .3398, approximately, around the circle from (1, 0), and the other point, because it is symmetrically placed, is that same distance from the point (-1, 0). Since (-1, 0) is the entire half circle, a distance of \pi from (1, 0), that symmetrical point is a distance \pi- .3398 from it, just as 5\pi/6= \pi- \pi/6.