What is the principal solutions of the equation \(\tan x=-\frac{1}{\sqrt{3}}\)?

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  1. \(\frac{9 \pi}{3}, \frac{7 \pi}{3}\)
  2. \(\frac{2 \pi}{3}, \frac{2 \pi}{3}\)
  3. \(\frac{3 \pi}{6}, \frac{2 \pi}{6}\)
  4. \(\frac{11 \pi}{6}, \frac{5 \pi}{6}\)

Answer (Detailed Solution Below)

Option 4 : \(\frac{11 \pi}{6}, \frac{5 \pi}{6}\)
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Detailed Solution

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Concept:

 

The principal solutions of a trigonometric equation are those solutions that lie between 0 and 2π.

Formula:

General solution of tan(x) = tan(α) is  given as;

x = nπ + α where α ∈ (-π/2 , π/2) and n ∈ Z.

Calculation:

Given, \(\tan x=-\frac{1}{\sqrt{3}}\)

⇒ tan(x) = tan(-π/6)

∴ α = -π/6

⇒ x = nπ + (-π/6) , n ∈ Z

Putting n = 1 and 2, we get - 

x = 5π/6 and 11π/6

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