Find the general solution of the equation 2cos(3x)-1=0. Hence find the solutions for -pi < x < pi

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Math

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We wish to slove the problem for x

`2cos(3x)-1=0`

`2cos(3x)=1`

`cos(3x)=1/2`

`but`

`cos(pi/3)=1/2`

Thus

`cos(3x)=cos(pi/3)`

`3x=2npi+-pi/3`

`x=(2/3)npi+-pi/9`    , n is an integer.

It is general solution of the given problem.

If n=0

`x=+-pi/9 `

If n=1

`x=(2/3)pi+-pi/9`

`x=(7pi)/9 or (5pi)/9`

if n=2

`x=(4/3)pi+-pi/9`

`x=(13pi)/9 or (11pi)/9`  ,it is not in the given interval `(-pi,pi)`

Thus

`x=+-pi/9, (7pi)/9,(5pi)/9`

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