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Evaluate the value of the following $\cot {{120}^{\circ }}=?$

Last updated date: 22nd Jul 2024
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Hint: To solve the question, the concept of trigonometric value should be known. The values of trigonometric values for certain numbers should be known. Details of the trigonometric function is required to solve the question.

Complete step by step answer:
To start with some details on the trigonometric function, $\cot $. We know that the trigonometric function $\cot x$ is the reciprocal of the other trigonometric function $\tan x$, this could be mathematically represented as
$\cot x=\dfrac{1}{\tan x}$…………………………………………………………. (i)
On applying the same formula to find value of the given question,
$\Rightarrow \cot {{120}^{\circ }}=\dfrac{1}{\tan {{120}^{\circ }}}$
We know that the value of $\tan x$ for $x$ belonging from $0$ to ${{90}^{\circ }}$ and from ${{180}^{\circ }}$ to ${{270}^{\circ }}$are positive, and for rest of the values of $x$ between $0-{{360}^{\circ }}$ the value of $\tan x$will be negative.
The value of
$\Rightarrow \tan {{120}^{\circ }}$
The angle ${{120}^{\circ }}$ will be written as the complementary angle which is in terms as the sum of ${{90}^{\circ }}$ and ${{30}^{\circ }}$.
$\Rightarrow \tan \left( {{90}^{\circ }}+{{30}^{\circ }} \right)$
Since the function$\tan x$has angle in second quadrant the function will be negative:
$\Rightarrow -\cot {{30}^{\circ }}$
$\Rightarrow -\dfrac{1}{\sqrt{3}}$
On substituting the value of $\tan {{120}^{\circ }}$in the expression$\cot {{120}^{\circ }}=\dfrac{1}{\tan {{120}^{\circ }}}$, so doing this we get:
$\Rightarrow \cot {{120}^{\circ }}=\dfrac{1}{-\dfrac{1}{\sqrt{3}}}$
$\Rightarrow \cot {{120}^{\circ }}=-\sqrt{3}$

So, the correct answer is “Option D”.

Note: Calculation of the trigonometric function with a certain angle becomes much easier with the help of the graph. Minimum and maximum value of the function could easily be known to us with the help of a graph. The problem could directly be found by a shortcut method.
$\Rightarrow \cot {{120}^{\circ }}=\cot \left( {{90}^{\circ }}+{{30}^{\circ }} \right)$
Since the value of ${{120}^{\circ }}$ is in the second quadrant so the trigonometric function $\tan $ and $\cot $ will give the negative value, so on conversion it becomes:
$\Rightarrow - \tan {{30}^{\circ }}$
The value of $\tan {{30}^{\circ }}$ is $\sqrt{3}$ . So on substituting the value in the above expression we get:
$\Rightarrow - \sqrt{3}$