Cosx is the horizontal co-ordinate of a point on the circle of radius =1 if you like, centred at the origin.
When you obtain arccos using the value -0.5,
your calculator will return one of two angles this corresponds to.
You wrote your angles values "without" .
It's much simpler to approach this type of question,
by considering cosx to be a horizontal co-ordinate of value -0.5,
to the left of the origin.
This co-ordinate references 2 angles, one less than 180 degrees
and the other greater than 180 degrees.
If you calculate arccos(0.5) you obtain the acute angle that you add to or subtract from 180 degrees.
Note that the question excludes x=0 or x=360 degrees,
so it is asking you to find the 2 angles x, referenced by cosx=-0.5.