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

1 Answer
Dec 20, 2017

Answer:

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

Explanation:

#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\ne pi*n#, #n\inZZ#.