Vertical asymptotes are straight lines of the equation `x=a` , toward which a function `f(x) ` approaches infinitesimally closely, but never reaches the line, as `f(x)` increases without bound. For these values of x, the function is either unbounded or is undefined. For example, the function `x=a` has a vertical asymptote at `f(x)=1/x` , because the function is undefined there.
The cosecant function is the reciprocal of the sine function. So, whenever the sine function approaches 0, the cosecant function approaches the vertical limit of `(1/ sin theta) = (1/0) ` `=oo` . The sine function equals 0 at `theta` = `kpi` , where k is any positive or negative integer. These are also the vertical asymptotes of the cosecant function.