#csc x = 1/sin x#
#0.5 csc x = 0.5/sin x#
any number divided by #0# gives an undefined result, so #0.5# over #0# is always undefined.
the function #g(x)# will be undefined at any #x#-values for which #sin x = 0#.
from #0^@# to #360^@#, the #x#-values where #sin x = 0# are #0^@, 180^@ and 360^@#.
alternatively, in radians from #0# to #2pi#, the #x#-values where #sin x = 0# are #0, pi and 2pi#.
since the graph of #y = sin x# is periodic, the values for which #sin x = 0# repeat every #180^@, or pi# radians.
therefore, the points for which #1/sin x# and therefore #0.5/sin x# are undefined are #0^@, 180^@ and 360^@# (#0, pi and 2pi#) in the restricted domain, but can repeat every #180^@#, or every #pi# radians, in either direction.
graph{0.5 csc x [-16.08, 23.92, -6.42, 13.58]}
here, you can see the repeating points at which the graph cannot continue due to undefined values. for example, the #y#-value steeply increases when approaching closer to #x = 0# from the right, but never reaches #0#. the #y#-value steeply decreases when approaching closer to #x = 0# from the left, but never reaches #0#.
in summary, there are an infinite number of asymptotes for the graph #g(x) = 0.5 csc x#, unless the domain is restricted. the asymptotes have a period of #180^@# or #pi# radians.
