Cotangent Function cot x

# Cotangent Function cot x

## Definition and Graph of the Cotangent Function

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

Given point $M(x,y)$ on the terminal side, the cotangent function is defined as
$\cot(\theta) = \dfrac{x}{y}$
$\cot(\theta)$ may also be expressed in terms of $\sin(\theta)$ and $\cos(\theta)$ as follows:
$\cot(\theta) = \dfrac{x}{y} = \dfrac{x/r}{y/r} = \dfrac{\cos(\theta)}{\sin(\theta)}$

Note that
1) $\cot(\theta+\pi) = \dfrac{\cos(\theta+\pi)}{\sin(\theta+\pi)} = \dfrac{-\cos(\theta)}{-\sin(\theta)}= \dfrac{\cos(\theta)}{\sin(\theta)} = \cot(\theta)$
and we therefore conclude that $\cot(\theta)$ is a periodic function with a period equal to $\pi$.

2) $\cot(-\theta) = \dfrac{\cos(-\theta)}{\sin(-\theta)} = \dfrac{ \cos(\theta)}{-\sin(\theta)} = - \dfrac{\cos(\theta)}{\sin(\theta)} = - \cot(\theta)$
and therefore $\cot(\theta)$ is an odd function which also means that the graph of $\cot(\theta)$ is symmetric with respect to the origin of the system of coordinates.

We now use a unit circle to find $\sin(\theta)$ and $\cos(\theta)$ and hence $\cot(\theta)$ over one period extending from $\theta = 0$ to $\theta = \pi$.
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 angles $0, \dfrac{\pi}{4} , \dfrac{\pi}{2} , \dfrac{3\pi}{4} , \pi$ and the corresponding values of $\cos(\theta)$ and $\sin(\theta)$ on a table as shown below.

 $\theta$ $\cos(\theta)$ $\sin(\theta)$ $\cot(\theta) = \dfrac{\cos(\theta)}{\sin(\theta)}$ $0$ $1$ $0$ $undefined$ $\dfrac{\pi}{4}$ $\dfrac{\sqrt 2}{2}$ $\dfrac{\sqrt 2}{2}$ $1$ $\dfrac{\pi}{2}$ $0$ $1$ $0$ $\dfrac{3\pi}{4}$ $- \dfrac{\sqrt 2}{2}$ $\dfrac{\sqrt 2}{2}$ $- 1$ $\pi$ $- 1$ $0$ $undefined$

$\cot(\theta)$ is undefined at $\theta = 0$ and $\theta = \pi$, however we can get information about the behaviour $\cot(\theta)$ close to these values.
We use the calculator to find values of $\cot(\theta)$ as $\theta$ approaches $0$ starting at $\theta = -0.1$
 $\theta$ $\cot(\theta)$ $-0.1$ $-9.966644423$ $-0.01$ $-99.99666664$ $-0.001$ $-999.9996667$ $-0.000001$ $-1000000$

As $\theta$ approaches $0$ by values smaller than $0$, $\cot(\theta)$ approaches small values.
Let us now consider values that approaches 0 such that these values are larger than 0.
 $\theta$ $\cot(\theta)$ $0.1$ $9.966644423$ $0.01$ $99.99666664$ $0.001$ $999.9996667$ $0.000001$ $1000000$
As $\theta$ approaches $0$ by values larger than $0$ , $\cot(\theta)$ approaches large values and hence the existence of a vertical asymptote at $\theta = 0$.

Using the concept of limits, we describe the behaviour of $\cot(\theta)$ as $\theta$ approaches $0$ from the left (or by values smaller than $0$) as follows
$\lim_{\theta \to 0^-} \cot(\theta) = -\infty$
and the behaviour of $\cot(\theta)$ as $\theta$ approaches $0$ from the right (or by values larger than $0$) as follows
$\lim_{\theta \to 0^+} \cot(\theta) = +\infty$

We now use a system of rectangular axes $(x,y)$ to plot the points in the above table and approximate the graph of the cotangent cot x function as shown below.

NOTE
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 $\cot(\theta)$ which is labeled as $y = \cot(x)$.

## Properties of cot x

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

## Interactive Tutorial on the General Cotangent Function

The general cotangent function given by

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

and its period, phase shift, asymptotes domain and range are explored interactively. An app is used, where parameters a, b, c and d are changed to investigate their effects on the graph of f.

 a = 1 b = 1 c = 0 d = 0
>

1 - Set a = 1, b = 1, c = 0 and d = 0. Take note of the period, phase shift and positions of the asymptotes (vertical lines) of the graph of f? Now change a , how does it affect the graph? Does it affect its range? If yes, how?
2 - Set a = 1, c = 0, d = 0 and change b. Approximate the period from the graph and compare it to $\dfrac{2\pi}{| b |}$. How does b affect the graph of f(x)? How does it affect the asymptotes of the graph of f?
3 - Set a = 1, b = 1, d = 0 and change c starting from zero increasing slowly to positive large values. Take note of the shift, is it to the left or to the right? Compare its measure to $- c / b$.
4 - Set a = 1, b = 1, d = 0 and change c starting from zero deceasing slowly to negative smaller values. Take note of the shift, is it a left or a right shift? Compare its measure to $- c / b$.
5 - Set a,b and c to non zero values and change d. What is the direction of the shift of the graph?
6 - Which parameters affect the positions of the vertical asymptotes? Explain analytically.
7 - Which parameters affect the domain of the cotangent function? Explain analytically.
8 - Which parameters affect the range of the cotagent function? Explain analytically.

## More References and Links Related to the Cotangent cot x function

Sine Function sin x
Cosine Function cos x
Trigonometric Functions.