Question

principal solution of the equation \[\cot x = - \sqrt 3 \] is
A. \[\dfrac{\pi }{3}\]
B. \[\dfrac{{2\pi }}{3}\]
C. \[\dfrac{\pi }{6}\]
D. \[\dfrac{{5\pi }}{6}\]

Answer 1

Hint: As we know that the \[cot{\rm{ }}x\;\] is one of the trigonometric identities (other trigonometric identities are sin x, cos x, tan x, sec x, cosec x, where the x could be any angle from 0 to\[360^\circ \]). We can find the values of different angles of the different trigonometric identities in the trigonometric table. We should be remembering that the principal solution is the solution that lies between these angles. So the answer will definitely lie between the 0 and \[2\pi \].


Complete step by step solution
Given:
The equation is \[\cot x = - \sqrt 3 \].
On rearranging the equation, we get,
\[x = {\cot ^{ - 1}}\left( { - \sqrt 3 } \right)\]
As we know that, according to the trigonometry table, \[\cot 30^\circ \] or \[\cot \dfrac{\pi }{6} = \sqrt 3 \], so placing \[\cot \left( { - \dfrac{\pi }{6}} \right)\] in\[\sqrt 3 \]. Then, the equation can be written as:
 \[x = {\cot ^{ - 1}}\left( {\cot \left( { - \dfrac{\pi }{6}} \right)} \right)\]
Now, we know that the principal solution of the \[co{t^{ - 1}}{\rm{ }}x\] lies between 0 and \[\pi \], so we are assuming that the value of x lies between 0 and \[\pi \], then the equation can be written as:
\[\begin{array}{c}
x = \pi - \dfrac{\pi }{6}\\
x = \dfrac{{5\pi }}{6}
\end{array}\]
If we check the output, we can say that \[\dfrac{{5\pi }}{6}\] lies between 0 and\[\pi \], so the result is correct.
Therefore, the principal solution of the \[\cot x = - \sqrt 3 \] is \[\dfrac{{5\pi }}{6}\], it means the option (d) is correct.


Note: To solve this problem, we have to be aware about the trigonometric table, a trigonometric table is an output value of every trigonometric identity according to the angle that we substitute. For example, the value of \[\cot 30^\circ \] is \[\sqrt 3 \], like so the value of \[\cot 45^\circ \] can be found through it. It should be noted that the values of cot are exactly opposite the value of tan in the trigonometric table.