For a ratio #r#, #arctan(r) = theta#
#color(white)("XXXX")#where #theta epsilon [-pi/2, pi/2]#
and
#color(white)("XXXX")##tan(theta) = r#
#color(white)("XXXX")##color(white)("XXXX")##color(white)("XXXX")#(definition)
The question therefore becomes:
#color(white)("XXXX")#For what angle #theta epsilon [-pi/2, pi/2]# is
#color(white)("XXXX")##color(white)("XXXX")##color(white)("XXXX")##tan(theta) = tan((-2pi)/3)# ?
Note that the angle #((-2pi)/3)# occurs in the third quadrant and is a positive value (based on CAST or recognition that both the "rise" and the "run" are negative, so the ratio #tan = rise/run# is positive).
The required angle #theta# must occur in the first quadrant (since #theta epsilon [-pi/2, pi,2]# and #tan(theta) >= 0#)
Note that the reference angle (see diagram above) for #(-(2pi)/3)# is #pi/3#.
