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 $$\left\{x:x\in R\mathrm{a}\mathrm{n}\mathrm{d}x\ne n\pi ,n\in Z\right\}$$. 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 $$x=-{\displaystyle \frac{3\pi }{2}}$$ in the function $$y=\cot x$$, we get
$$\begin{array}{l}y=\cot \left(-{\displaystyle \frac{3\pi }{2}}\right)\\ \Rightarrow y=0\end{array}$$
Substituting $$x=-{\displaystyle \frac{\pi }{2}}$$ in the function $$y=\cot x$$, we get
$$\begin{array}{l}y=\cot \left(-{\displaystyle \frac{\pi }{2}}\right)\\ \Rightarrow y=0\end{array}$$
Substituting $$x={\displaystyle \frac{\pi }{2}}$$ in the function $$y=\cot x$$, we get
$$\begin{array}{l}y=\cot \left({\displaystyle \frac{\pi }{2}}\right)\\ \Rightarrow y=0\end{array}$$
Substituting $$x={\displaystyle \frac{3\pi }{2}}$$ in the function $$y=\cot x$$, we get
$$\begin{array}{l}y=\cot \left({\displaystyle \frac{3\pi }{2}}\right)\\ \Rightarrow y=0\end{array}$$
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

$$x$$	$$y$$
$$-2\pi$$	$$\mathrm{\infty }$$
$$-{\displaystyle \frac{3\pi }{2}}$$	$$0$$
$$-\pi$$	$$\mathrm{\infty }$$
$$-{\displaystyle \frac{\pi }{2}}$$	$$0$$
$$0$$	$$\mathrm{\infty }$$
$${\displaystyle \frac{\pi }{2}}$$	$$0$$
$$\pi$$	$$\mathrm{\infty }$$
$${\displaystyle \frac{3\pi }{2}}$$	$$0$$
$$2\pi$$	$$\mathrm{\infty }$$

The value of $$y=\cot x$$ decreases from $$\mathrm{\infty }$$ to 0 at $$x=-{\displaystyle \frac{3\pi }{2}}$$, and then to $$-\mathrm{\infty }$$ in the interval $$\left(-2\pi ,-\pi \right)$$.
Similarly, the value of $$y=\cot x$$ decreases from $$\mathrm{\infty }$$ to 0 at $$x=-{\displaystyle \frac{\pi }{2}},{\displaystyle \frac{\pi }{2}},{\displaystyle \frac{3\pi }{2}}$$, and then to $$-\mathrm{\infty }$$ in the intervals $$\left(-\pi ,0\right)$$, $$\left(0,\pi \right)$$, and $$\left(\pi ,2\pi \right)$$.
Now, we will use the points $$\left(-{\displaystyle \frac{3\pi }{2}},0\right)$$, $$\left(-{\displaystyle \frac{\pi }{2}},0\right)$$, $$\left({\displaystyle \frac{\pi }{2}},0\right)$$, $$\left({\displaystyle \frac{3\pi }{2}},0\right)$$ 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 $$-\mathrm{\infty }$$ to $$\mathrm{\infty }$$. As tangent function is a reciprocal function cotangent function, so their graph faces opposite to each other.