Question

How do you solve the following equation $\cot \left( {\dfrac{x}{2}} \right) = 1$ in the interval $[0,2\pi ]$ ?

Answer 1

Hint : The general solution of the trigonometric equation $\tan x = \tan y$ is $x = n\pi + y$ where $n \in \mathbb{Z}$ . Arrange the given equation in the above form and use the result to get the general solution.

Complete step-by-step answer :
The equation contains the trigonometric function $\cot x$ . So, we can here try to use the known general solution for the equation $\tan x = \tan y$ and the result $\cot x = \dfrac{1}{{\tan x}}$ . We will first try to write the equation in this form.
We know that $\cot x = \dfrac{1}{{\tan x}}$ . So, the given equation becomes, $\dfrac{1}{{\tan \left( {\dfrac{x}{2}} \right)}} = 1$ .
$\Rightarrow \tan \left( {\dfrac{x}{2}} \right) = 1$
We also know that $\tan \left( {\dfrac{\pi }{4}} \right) = 1$ .
$\Rightarrow \tan \left( {\dfrac{x}{2}} \right) = \tan \left( {\dfrac{\pi }{4}} \right)$
Now, the equation is of the form $\tan x = \tan y$ and we know that the solution of such an equation is $x = n\pi + y$ where $n \in \mathbb{Z}$ .
$\Rightarrow \dfrac{x}{2} = n\pi + \dfrac{\pi }{4}$
$\Rightarrow x = 2n\pi + \dfrac{\pi }{2}$ which is the general solution of the given trigonometric equation.
But to find the solution that lies in the interval $[0,2\pi ]$ we will look at the solutions generated by the general solution. Here we consider the solution generated by $n = 0,1,2,...$ Because we are interested in finding the solution in the positive interval $[0,2\pi ]$ .
The general solution gives us,
$x = 2(0)\pi + \dfrac{\pi }{2},{\text{ }}2(1)\pi + \dfrac{\pi }{2},{\text{ }}2(2)\pi + \dfrac{\pi }{2}...$ for $n = 0,1,2,...$
$\Rightarrow x = \dfrac{\pi }{2},{\text{ }}2\pi + \dfrac{\pi }{2},{\text{ 4}}\pi + \dfrac{\pi }{2}...$
$\Rightarrow x = \dfrac{\pi }{2},{\text{ }}\dfrac{{5\pi }}{2},{\text{ }}\dfrac{{9\pi }}{2}...$
Here, $x = \dfrac{\pi }{2} \in [0,2\pi ]$ . So, it is the only solution in the interval $[0,2\pi ]$ for the equation $\cot \left( {\dfrac{x}{2}} \right) = 1$ .
Additional information:
Other important general solutions to solve the trigonometric equations are,
$\sin x = \sin y \Rightarrow x = n\pi + {( - 1)^n}y,{\text{ where }}n \in \mathbb{Z}$
$\cos x = \cos y \Rightarrow x = 2n\pi \pm y,{\text{ where }}n \in \mathbb{Z}$

Note : It is very important to keep the interval in mind while solving the equation. Because that itself would make the steps simpler and can help in choosing the required solution.