Correct Answer - Option 3 : 11π/6
CONCEPT:
The general solution of the equation tan x = tan α is given by: x = nπ + α, where \(\alpha ∈ \left( { - \frac{π }{2},\frac{π }{2}} \right)\) and n ∈ Z
Note: The solutions of a trigonometric equation for which 0 ≤ x < 2π are called principal solution.
CALCULATION:
Given: \(\tan \ x = -\frac{1}{\sqrt 3} \)
As we know that, \(\tan \ \frac{5π}{6} = -\frac{1}{\sqrt 3} \)
⇒ \(\tan \ x = \tan \ \frac{5π}{6}\)
As we know that, if tan x = tan α then x = nπ + α, where \(\alpha ∈ \left( { - \frac{π }{2},\frac{π }{2}} \right)\) and n ∈ Z
⇒ x = nπ + (5π/6) where n ∈ Z.
As we know that, if the solutions of a trigonometric equation for which 0 ≤ x < 2π are called principal solution.
So, the principal solutions of the given equation are x = 5π/6 and 11π/6
Hence, the correct option is 3.
