Question

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$\bf\Huge\red{\mid{\overline{\underline{ QUESTION }}}\mid }$

$\blue{write \: the \: principal \: solution \: of \: \\ the \: equarion}$

$\Large\mathfrak\blue{\implies \: TAN X = - \frac{1}{ \sqrt{3} }}$

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Answer 1

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1. A solution of a trigonometric equation is the value of the unknown angle that satisfies the equation.
2. Consider the equation sin θ = ½.
3. Clearly we see that principal solution of the equation sin θ = ½ are π/6 and 5π/6 because these solutions lie between 0 to 2π.

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Answer 2

Answer:

x = 5π/6 , 11π/6

Note:

★ The possible values of the variable which satisfy the given equation are called its solutions or roots.

★ Principal solution of a trigonometric equation : The possible angles ( ie; the possible values of variable x ) in one rotation ( ie ; 0 ≤ x ≤ 2π ) which satisfy any trigonometric equation are called its principal solutions.

Solution:

Given: tanx = -1/√3

To find: Principal solutions

Now,

We know that ;

If tanx = 1/√3

then x = π/6 ( angle for the 1st quadrant)

Now ,

We shall use the angle π/6 and the horizontal axes (ie ; 0 or π or 2π) to obtain the principal solutions of the given trigonometric equation tanx = -1/√3 .

Also,

We know that , tanx is negative in the 2nd quadrant and the 4th quadrant .

Thus,

=> tanx = -1/√3

=> tanx = tan(π - π/6) or tan(2π - π/6)

=> tanx = tan(5π/6) or tan(11π/6)

=> x = 5π/6 or 11π/6

Where x = 5π/6 lies in 2nd quadrant and x = 11π/6 lies in 4th quadrant .

Hence,

The required principal solution are :

x = 5π/6 , 11π/6