# Cosecant Function csc x

## Definition and Graph of the Coseacnt Function

Angle $\theta$ with initial side on the positive x axis and terminal side OM is shown below. It is an angle in standard position.

The cosecant function $csc (\theta)$ is defined as
$\csc(\theta) = \dfrac{r}{y}$ , where $r$ is the distance from O to M and is given by $r = \sqrt{x^2+y^2}$.
An examination of the definiton of the cosecant gives a relationship between $csc (\theta)$ and $sin (\theta)$ as follows
$\csc(\theta) = \dfrac{r}{y} = \dfrac{1}{\sin(\theta)}$

Note that
1) $\csc(\theta+2\pi) = \dfrac{1}{\sin(\theta+2\pi)} = \dfrac{1}{\sin(\theta)}= \csc(\theta)$
and therefore $\csc(\theta)$ is a periodic function whose period is equal to $2\pi$.

2) $\csc(-\theta) = \dfrac{1}{\sin(-\theta)} = \dfrac{1}{-\sin(\theta)} = - \dfrac{1}{\sin(\theta)} = - \csc(\theta)$
and therefore $\csc(\theta)$ is an odd function and its graph is symmetric with respect to the origin of a rectangular system of coordinates.

We now use a unit circle to find $\sin(\theta)$ and hence $\csc(\theta)$ over one period extending from $\theta = 0$ to $\theta = 2\pi$.
We know from the sine and cosine functions that the x and y coordinates on a unit circle gives the values of $\sin(\theta)$ and $\cos(\theta)$ as shown below.

Let us now put the values of the quadrantal angles angles $0, \dfrac{\pi}{2} , \pi , \dfrac{3\pi}{2} , 2\pi$ and the corresponding values of $\sin(\theta)$ and $\csc (\theta) = \dfrac{1}{\sin (\theta)}$ on a table as shown below.

 $\theta$ $\sin(\theta)$ $\csc (\theta) = \dfrac{1}{\sin (\theta)}$ $0$ $0$ undefined $\dfrac{\pi}{2}$ $1$ 1 $\pi$ $0$ undefined $\dfrac{3\pi}{2}$ $-1$ -1 $2\pi$ $0$ undefined

$\csc(\theta)$ is undefined at $\theta = 0$, $\theta = \pi$ and $\theta = 2\pi$ , however we can get information about the behaviour $\csc(\theta)$ close to these values using a claculator.
We use the calculator to find values of $\csc(\theta)$ as $\theta$ approaches $0$ starting at $\theta = -0.1$
 $\theta$ $\csc(\theta)$ $-0.1$ $-10.01668613$ $-0.01$ $-100.0016667$ $-0.001$ $-1000.000167$ $-0.000001$ $-1000000$

As $\theta$ approaches $0$ by values smaller than $0$, $\csc(\theta)$ approaches small values.

We now use the calculator to find values of $\csc(\theta)$ as $\theta$ approaches $0$ starting at $\theta = 0.1$
 $\theta$ $\csc(\theta)$ $0.1$ $10.01668613$ $0.01$ $100.0016667$ $0.001$ $1000.000167$ $0.000001$ $1000000$
Similarly, as $\theta$ approaches $0$ by values larger than $0$ , $\csc(\theta)$ approaches large values and hence the existence of a vertical asymptote at $\theta = 0$.

Using the concept of limits, we describe the behaviour of $\csc(\theta)$ as $\theta$ approaches $0$ from the left (or by values smaller than $0$) as follows
$\lim_{\theta \to 0^-} \csc(\theta) = -\infty$
and the behaviour of $\csc(\theta)$ as $\theta$ approaches $0$ from the right (or by values larger than $0$) as follows
$\lim_{\theta \to 0^+} \csc(\theta) = +\infty$
Similar behaviour occur close to all values of $x = n\pi$ where $n$ is an integer.

We now use a system of rectangular coordinates $(x,y)$ to plot the points in the above table and approximate the graph of the cosecant function $\csc x$ as shown below.
$\csc (\theta)$ from $\theta = 0$ to $\theta = 2\pi$ is shown in red along with $\sin (\theta)$ in green to understand the relationship, of the two functions over one period, graphically. The most important observation is: the vertical asymptotes (shown in broken lines) of $\csc (\theta)$ occur at the position of the zeros of $\sin (\theta)$.

NOTE
Because we are used to $x$ being the variable of a function, $x$ on the graph takes values of $\theta$ and y takes the values of $\csc(\theta)$ which is noted as $y = \csc(x)$. The broken vertical lines indicate the vertical asymptotes of $\csc x$.

## Properties of csc x

1) csc x has a period equal to $2\pi$.
2) $\csc(x)$ has vertical asymptotes at all values of $x = n\pi$ , $n$ being any integer.
3) The domain of $\csc(x)$ is the set of all real numbers except $x = n\pi$ , $n$ being any integer.
4) The range of $\csc(x)$ is given by: $(-\infty , -1] \cup [1, +\infty)$
5) $\csc(x)$ is an odd function and its graph is symmetric with respect to the origin of the system of axes.

## Interactive Tutorial on the Cosecant Function csc x of the General Form

A tutorial on exploring the cosecant function of the general form given by

$f(x) = a \csc( b x + c ) + d$

is presented. An interactive app is used where parameters a, b, c and d may be changed and their effects on the period, phase shift, asymptotes, domain and range explored.
Period is given by: $\dfrac{2\pi}{|b|}$
Use the app to answer the questions below.
 a = 1 b = 1 c = 0 d = 0
>

1 - Use the scroll bar to set a = 1, b = 1, c = 0 and d = 0. Now change a , how does it affect the graph? Does it affect the range? If yes how?

2 - Set a = 1, c = 0, d = 0 and change b. For each value of b find the period from the graph and compare it to $\dfrac{2\pi}{|b|}$, formula of the period of $f(x) = a \csc( b x + c ) + d$. How does b affect the graph of f(x)? How does it affect the vertical asymptotes?

3 - Set a = 1, b = 1, d = 0 and change c starting from zero going slowly to positive large values. Take note of the shift, is it left or right, and compare it to $- c / b$, formula of the phase shift of $f(x) = a \csc( b x + c ) + d$.

4 - Set a = 1, b = 1, d = 0 and change c starting from zero going slowly to negative smaller values. Take note of the shift, is it left or right, and compare it to $- c / b$.

5 - Repeat steps 4 and 5 above for b = 2, 3 and 4.

6 - Set a, b and c to non zero values and change d. What is the direction of the shift of the graph? How is the range of the function affected?

7 - Which of the parameters affect the positions of the vertical asymptotes? Explain analytically.

8 - Which of the parameters affect the domain of the function? Explain analytically.

9 - Which of the parameters affect the range of the cosecant function? Explain analytically.

## More references on the trigonometric functions

Properties of Trigonometric Functions.
Trigonometric Functions.