What is the domain and range of #y=csc x#?

Question

Hazel Anglin · Accepted Answer

Domain of #y=csc(x)# is #x\inRR, x e pi*n#, #n\inZZ#.
Range of #y=csc(x)# is #y<=-1# or #y>=1#.

#y=csc(x)# is the reciprocal of #y=sin(x)# so its domain and range are related to sine's domain and range. Since the range of #y=sin(x)# is #-1<=y<=1# we get that the range of #y=csc(x)# is #y<=-1# or #y>=1#, which encompasses the reciprocal of every value in the range of sine. The domain of #y=csc(x)# is every value in the domain of sine with the exception of where #sin(x)=0#, since the reciprocal of 0 is undefined. So we solve #sin(x)=0# and get #x=0+pi*n# where #n\inZZ#. That means the domain of #y=csc(x)# is #x\inRR, x e pi*n#, #n\inZZ#.

Genesis Allen · Answer

undefined The domain of the cosecant function, $ y = \csc(x) $, consists of all real numbers except for the values where the sine function is zero, because division by zero is undefined. The sine function is zero at integer multiples of $ \pi $, so the domain of $ y = \csc(x) $ is $ x \neq n\pi $, where $ n $ is an integer.

The range of $ y = \csc(x) $ is the set of all real numbers excluding zero, since the cosecant function has vertical asymptotes at its zeros. So, the range is $ y \in (-\infty, -1] \cup [1, \infty) $.

Julia Adams · Answer

undefined The domain of y=csc(x) is all real numbers except where sin(x) = 0, which occurs at x = nπ where n is an integer. The range of y=csc(x) is all real numbers except y = 0.