Question

The principal value of ${\cot ^{ - 1}}\left( { - 1} \right)$ is:-
A.$\dfrac{\pi }{4}$
B.\[ - \dfrac{\pi }{4}\]
C.$\dfrac{{3\pi }}{4}$
D.None of these

Answer 1

Hint: We will first write ${\cot ^{ - 1}}\left( { - 1} \right) = x$. Then, find the angle where \[\cot x = 1\]. Now, as we know that cot can be negative in the second and fourth quadrant only, find their corresponding values in those quadrants. The values in the cartesian plane between $0 \leqslant x < 2\pi $ such that $\cot x = - 1$ will be the principal values of ${\cot ^{ - 1}}\left( { - 1} \right) = x$.

Complete step-by-step answer:
We have to find the principal value of ${\cot ^{ - 1}}\left( { - 1} \right)$. That is, we want to find the angle where $\cot x = - 1$ in the cartesian plane such that $0 \leqslant x < 2\pi $
We will know let ${\cot ^{ - 1}}\left( { - 1} \right) = x$, which implies that $\cot x = - 1$
Since, the value is negative, and cot of any angle is negative in the second and fourth quadrant only.
Hence, the value of $x$ is in the second and fourth quadrant.
We also know that \[\cot \left( {\dfrac{\pi }{4}} \right) = 1\]
Therefore, we will have the corresponding angle in the second and fourth quadrant.
Let us first find the value in the second quadrant.
For, writing the angle in second quadrant whose value will be negative of \[\cot \left( {\dfrac{\pi }{4}} \right)\] , we will subtract $\dfrac{\pi }{4}$ from $\pi $.
Hence, we have
 $
  \cot \left( {\pi - \dfrac{\pi }{4}} \right) = - 1 \\
  \cot \left( {\dfrac{{3\pi }}{4}} \right) = - 1 \\
$
Hence, the principal value is \[\dfrac{{3\pi }}{4}\] in the second quadrant.
Now, we will find the value in the fourth quadrant.
For, writing the angle in second quadrant whose value will be negative of \[\cot \left( {\dfrac{\pi }{4}} \right)\] , we will subtract $\dfrac{\pi }{4}$ from $2\pi $.
Hence, we have
 $
  \cot \left( {2\pi - \dfrac{\pi }{4}} \right) = - 1 \\
   \Rightarrow \cot \left( {\dfrac{{7\pi }}{4}} \right) = - 1 \\
$
Hence, the principal value is \[\dfrac{{7\pi }}{4}\] in the fourth quadrant.
Therefore, option C is correct.


Note: A cartesian plane is divided into four quadrants. All trigonometry ratios are positive in the first quadrant, in the second quadrant only sin and cosec are positive, in third quadrant only ratios which are positive are of tan and cot, in fourth quadrant cos and secant will give positive answers.