# How do you find the domain & range for y=cscx?

May 19, 2016

#### Explanation:

Cosecant function is closely tied to sine function, as it is its reciprocal. If we relate to coordinate plane, while sine function is $\frac{y}{r}$, cosecant function is $\frac{r}{y}$ and problem arises when $y \to 0$, which happens when angle is $0$ or $\pi$ or $2 \pi$, $3 \pi$, etc.

Hence, domain of $\csc x$ is given by

$x \ne n \pi$, where $n$ is an integer.

Again as cosecant function is $\frac{r}{y}$ and as $r$ in $\frac{r}{y}$ is always positive and $r > y$, this function is always greater than or equal to $1$ or less than or equal to $- 1$.

Tis means, it never takes values between $1$ and $- 1$.

Further, it is equal to $1$ when $x = 2 n \pi + \frac{\pi}{2}$ and is equal to $- 1$ when $x = 2 n \pi - \frac{\pi}{2}$, where $n$ is an integer.

The graph of $\csc x$ will appear as shown below.

graph{cscx [-10, 10, -5, 5]}