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.
