Question

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

Answer 1

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`.