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

Dec 20, 2017

Domain of $y = \csc \left(x\right)$ is $x \setminus \in \mathbb{R} , x \setminus \ne \pi \cdot n$, $n \setminus \in \mathbb{Z}$.
Range of $y = \csc \left(x\right)$ is $y \le - 1$ or $y \ge 1$.
$y = \csc \left(x\right)$ is the reciprocal of $y = \sin \left(x\right)$ so its domain and range are related to sine's domain and range.
Since the range of $y = \sin \left(x\right)$ is $- 1 \le y \le 1$ we get that the range of $y = \csc \left(x\right)$ is $y \le - 1$ or $y \ge 1$, which encompasses the reciprocal of every value in the range of sine.
The domain of $y = \csc \left(x\right)$ is every value in the domain of sine with the exception of where $\sin \left(x\right) = 0$, since the reciprocal of 0 is undefined. So we solve $\sin \left(x\right) = 0$ and get $x = 0 + \pi \cdot n$ where $n \setminus \in \mathbb{Z}$. That means the domain of $y = \csc \left(x\right)$ is $x \setminus \in \mathbb{R} , x \setminus \ne \pi \cdot n$, $n \setminus \in \mathbb{Z}$.