Evaluate following sin 2 ?/3 and sin (-2 ?/3)
Solution: The first evaluation in this part uses the angle 2 ?/3. It is not on our unit circle above, though notice that 2 ?/3= ? - ?/3 . thus 2 ?/3 is found through rotating up from the -ve x-axis.
It means that the line for 2 ?/3 will be a mirror image of the line for ?/3 only within the second quadrant. The coordinates for 2 ?/3 will be the coordinates for ?/3 except the x coordinate will be negative.
Similarly for - 2 ?/3 we can notice that - 2 ?/3 = - ?+ ?/3 , hence this angle can be found by rotating down ?/3 from the -ve x-axis. It means that the line for ?/3 will be a mirror image of- 2 ?/3 the line for ?/3 only in the third quadrant and the coordinates will be the same as the coordinates for ?/3 except both will be negative.
Both of these angles along with their coordinates are illustrated on the following unit circle.
From this unit circle we can see that sin ( 2 ?/3 ) =√3/2 and sin ( -2 ?/3 )=√3/2
It leads to a nice fact regarding the sine function. The sine function is known an odd function and hence for any angle we have
sin ( - θ ) = - sin (θ )