How do i simplify 4sin^2 (3x)-3=0 in order to get the fundamental solution set?

2 Answers

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gsenviro | College Teacher | (Level 1) Educator Emeritus

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4 sin^2(3x) -3=0 can be written as

4 sin^2(3x) = 3

or, sin^2(3x) = 3/4

or, sin 3x = `+-sqrt(3)/2`  

now, sine function has a value of `sqrt(3)/2`  for `pi`/3, 2`pi`/3 in every 2`pi` interval.

i.e., 3x = `pi/3`  , `(2pi)/3`,`(7pi)/3` , `(8pi)/3` ,........

solution set is: 3x1 = `pi/3 + 2npi` and 3x2 = `(2pi)/3 + 2npi`  

or, the solution set is x1 = `pi/9 + (2npi)/3`  and x2= `(2pi)/9 + (2npi)/3`

Similarly, a sine function has a value of `-sqrt(3)/2`  for `(4pi)/3, (5pi)/3, (10pi)/3, (11pi)/3,....`

and the solution set would be:

3 x3 = `(4pi)/3 +2npi`   and 3 x4 = `(5pi)/3 +2npi`

i.e., the solution set is x3 = `(4pi)/9 + (2npi)/3`  and x4= `(5pi)/9 + (2npi)/3`

The fundamental solution set is x1, x2, x3 and x4.

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