Chapter 4.6, Problem 51E

To determine

To find: the solutions of the equation in the interval $\left[-2\pi ,2\pi \right]$ .

Answer to Problem 51E

  $x=-\left(\frac{4\pi }{3}\right),-\left(\frac{\pi }{3}\right),\left(\frac{2\pi }{3}\right)$ and $\left(\frac{5\pi }{3}\right)$

Explanation of Solution

Given information:

Given function is

  $\cot x=-\frac{\sqrt{3}}{3}$ .

Calculation:

Let us consider the points to solve the equation $\cot x=-\frac{\sqrt{3}}{3}$ .

Let us, first plot the graph for $f\left(x\right)=\cot x$

Now, in the above graph, look for all the points on the graph where the value of $\cot x$ is $-\frac{\sqrt{3}}{3}$ .

From the above graph, it can be seen that the value of $\tan x$ equal to $-\frac{\sqrt{3}}{3}$ at

  $x=-\left(\frac{4\pi }{3}\right),-\left(\frac{\pi }{3}\right),\left(\frac{2\pi }{3}\right)$ and $\left(\frac{5\pi }{3}\right)$

Therefore,

  $\cot x=-\frac{\sqrt{3}}{3}\text{a}tx=-\left(\frac{4\pi }{3}\right),-\left(\frac{\pi }{3}\right),\left(\frac{2\pi }{3}\right)$ and $\left(\frac{5\pi }{3}\right)$ .