Question

How do you graph \[y = \cot x\]?

Answer 1

Hint: We need to graph the given function. We will use the domain and some values of \[x\] lying between \[ - 2\pi \] and \[2\pi \] to find some values of \[y\]. Then, we will observe the behavior of the value of \[y\], and use it and the coordinates obtained to graph the function.

Complete step-by-step solution:
The domain of the function \[y = \cot x\] is given by \[\left\{ {x:x \in R{\rm{ and }}x \ne n\pi ,n \in Z} \right\}\]. This means that the cotangent of any multiple of \[\pi \] does not exist.
The graph of the cotangent function reaches arbitrarily large positive or negative values at these multiples of \[\pi \].
Now, we will find some values of \[y\] for some values of \[x\] lying between \[ - 2\pi \] and \[2\pi \].
Substituting \[x = - \dfrac{{3\pi }}{2}\] in the function \[y = \cot x\], we get
\[\begin{array}{l}y = \cot \left( { - \dfrac{{3\pi }}{2}} \right)\\ \Rightarrow y = 0\end{array}\]
Substituting \[x = - \dfrac{\pi }{2}\] in the function \[y = \cot x\], we get
\[\begin{array}{l}y = \cot \left( { - \dfrac{\pi }{2}} \right)\\ \Rightarrow y = 0\end{array}\]
Substituting \[x = \dfrac{\pi }{2}\] in the function \[y = \cot x\], we get
\[\begin{array}{l}y = \cot \left( {\dfrac{\pi }{2}} \right)\\ \Rightarrow y = 0\end{array}\]
Substituting \[x = \dfrac{{3\pi }}{2}\] in the function \[y = \cot x\], we get
\[\begin{array}{l}y = \cot \left( {\dfrac{{3\pi }}{2}} \right)\\ \Rightarrow y = 0\end{array}\]
The value of \[y\] at \[x = 2\pi ,\pi ,0,\pi ,2\pi \] is infinite.
Arranging the values of \[x\] and \[y\] in a table and writing the coordinates, we get

\[x\]	\[y\]
\[ - 2\pi \]	\[\infty \]
\[ - \dfrac{{3\pi }}{2}\]	\[0\]
\[ - \pi \]	\[\infty \]
\[ - \dfrac{\pi }{2}\]	\[0\]
\[0\]	\[\infty \]
\[\dfrac{\pi }{2}\]	\[0\]
\[\pi \]	\[\infty \]
\[\dfrac{{3\pi }}{2}\]	\[0\]
\[2\pi \]	\[\infty \]

The value of \[y = \cot x\] decreases from \[\infty \] to 0 at \[x = - \dfrac{{3\pi }}{2}\], and then to \[ - \infty \] in the interval \[\left( { - 2\pi , - \pi } \right)\].
Similarly, the value of \[y = \cot x\] decreases from \[\infty \] to 0 at \[x = - \dfrac{\pi }{2},\dfrac{\pi }{2},\dfrac{{3\pi }}{2}\], and then to \[ - \infty \] in the intervals \[\left( { - \pi ,0} \right)\], \[\left( {0,\pi } \right)\], and \[\left( {\pi ,2\pi } \right)\].
Now, we will use the points \[\left( { - \dfrac{{3\pi }}{2},0} \right)\], \[\left( { - \dfrac{\pi }{2},0} \right)\], \[\left( {\dfrac{\pi }{2},0} \right)\], \[\left( {\dfrac{{3\pi }}{2},0} \right)\] and the behaviour of the value of \[y = \cot x\] to graph the function.
Therefore, we get the graph

This is the required graph of the function \[y = \cot x\].

Note:
The period of the function \[y = \cot x\] is \[\pi \]. This means that the graph of \[y = \cot x\] will repeat for every \[\pi \] distance on the \[x\]-axis. It can be observed that the pattern and shape of the graph of \[y = \cot x\] is the same from \[ - 2\pi \] to \[ - \pi \], from \[ - \pi \] to 0, from 0 to \[\pi \], and from \[\pi \] to \[2\pi \]. The range of cotangent functions is from \[ - \infty \] to \[\infty \]. As tangent function is a reciprocal function cotangent function, so their graph faces opposite to each other.