`3tan^3 (x) = tan(x)` Find all the solutions of the equation in the interval `0,2pi)`.

Textbook Question

Chapter 5, 5.3 - Problem 27 - Precalculus (3rd Edition, Ron Larson).
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gsarora17 | (Level 2) Associate Educator

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`3tan^3(x)=tan(x)`

`3tan^3(x)-tan(x)=0`

`tan(x)(3tan^2(x)-1)=0 =>`

`tan(x)=0 , (3tan^2(x)-1)=0`

General solutions for tan(x)=0 are,

`x=0+pin`

Solutions in the range `0<=x<=2pi`  are,

`x=0 , pi ,2pi`

solving `(3tan^2(x)-1)=0`

`3tan^2(x)=1`

`tan^2(x)=1/3`

`tan(x)=+-1/sqrt(3)`

General solutions for tan(x)=-1/`sqrt(3)` are,

`x=(5pi)/6+pin`

solutions in the range `0<=x<=2pi`  are

`x=(5pi)/6 , (11pi)/6`

General solutions for tan(x)=1/`sqrt(3)` are

`x=pi/6+pin`

solutions in the range `0<=x<=2pi`  are,

`x=pi/6 , (7pi)/6`

Therefore solutions are,

`x=0 , pi/6 , pi , (5pi)/6 , (7pi)/6 , (11pi)/6 , 2pi`

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