Question

: Find principal and general solution of the equation $ \cot x = - \sqrt 3 $

Answer 1

Hint: In the given problem, to find the principal solution of the given equation we will use the value of trigonometric function $ \tan x $ for particular angle and also we will use basic knowledge of signs of trigonometric functions. To find a general solution of the given equation, we will use the result which is given by $ \tan x = \tan y \Rightarrow x = n\pi + y $ where $ n $ is integer.

Complete step-by-step answer:
In this problem, we have to find the principal and general solution of the equation $ \cot x = - \sqrt 3 $ . We know that $ \tan x = \dfrac{1}{{\cot x}} $ . Hence, we can write $ \tan x = - \dfrac{1}{{\sqrt 3 }} \cdots \cdots \left( 1 \right) $ .
In the equation $ \left( 1 \right) $ the value of $ \tan x $ is negative. Also we know that the trigonometric function $ \tan x $ is negative in the second and fourth quadrant. So, we can say that $ x $ will be in the second and fourth quadrant. Also we know that $ \tan x = \dfrac{1}{{\sqrt 3 }} \Rightarrow x = \dfrac{\pi }{6} $ .
In the second quadrant, we can say that
 $
  x = \pi - \dfrac{\pi }{6} \\
   \Rightarrow x = \dfrac{{5\pi }}{6} \;
  $
In the fourth quadrant, we can say that
 $
  x = 2\pi - \dfrac{\pi }{6} \\
   \Rightarrow x = \dfrac{{11\pi }}{6} \;
  $
Hence, principal solutions of the equation $ \cot x = - \sqrt 3 $ are $ x = \dfrac{{5\pi }}{6} $ and $ x = \dfrac{{11\pi }}{6} $ .
Now from the equation $ \left( 1 \right) $ , we can write $ \tan x = \tan \dfrac{{5\pi }}{6} \cdots \cdots \left( 2 \right) $ . We know that $ \tan x = \tan y \Rightarrow x = n\pi + y $ where $ n $ is integer. Use this information in equation $ \left( 2 \right) $ . So, we can write $ x = n\pi + \dfrac{{5\pi }}{6} $ where $ n $ is integer.
Hence, the general solution of the equation $ \cot x = - \sqrt 3 $ is $ x = n\pi + \dfrac{{5\pi }}{6} $ where $ n $ is integer.

Note: The solutions of a trigonometric equation for which $ 0 \leqslant x \leqslant 2\pi $ are called principal solutions. The mathematical expression involving integer $ n $ which gives all solutions of a trigonometric equation is called the general solution. In this type of problem, we must remember the values of trigonometric functions for particular angles. Also we must remember some trigonometric identities, results and formulas.