If #theta = cot^(-1)(-1/sqrt(3))#

the

the ratio #("opposite side")/("adjacent side")# of the defining triangle

must be

#color(white)("XXXX")##(-1):(sqrt(3))# (or equivalently #(1):(-sqrt(3))#)

We therefore have the conditions below:

These are standard reference triangles and based on their quadrants the corresponding angles (within the range #[0,360^o]#

are

#color(white)("XXXX")##180^o - 30^0 = 150^o = (5pi)/6# radians

and

#color(white)("XXXX")##360^o - 30^o = 330^o = (11pi)/6# radians

These can be combined and include angles outside the #[0,2pi]# range as

#color(white)("XXXX")##theta = 150^o +n*180^o#

or

#color(white)("XXXX")##theta = (5pi)/6 + n*pi#

#color(white)("XXXX")##color(white)("XXXX")##color(white)("XXXX")#for #AA n in ZZ#