Question

Solve 2sin(2x) - 2sinx +2sqrt3 cosx - sqrt3 = 0

Answer 1

x = +/-pi/3 or x = -pi/3 or (4/3)pi

Explanation:

2sin(2x)-2sinx+2sqrt3cosx-sqrt3 = 0

4sinxcosx - 2sinx + 2sqrt(3)cosx - sqrt(3) = 0

2sin(x)[2cos(x)-1] + sqrt(3)[2cos(x)-1] = 0

[2cos(x)-1][2sin(x)+sqrt(3)] = 0

2cos(x)-1 = 0 or 2sin(x)+sqrt(3) = 0

cos(x) = 1/2 or sin(x) = -sqrt(3)/2

x = +/-pi/3 or x = -pi/3 or (4/3)pi

Answer 2

2sin(2x)-2sinx+2sqrt3cosx-sqrt3 = 0

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4sin(x)cos(x) - 2sin(x) + 2sqrt(3)cos(x) - sqrt(3) = 0

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Factor:

2sin(x)[2cos(x)-1] + sqrt(3)[2cos(x)-1] = 0

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[2cos(x)-1][2sin(x)+sqrt(3)] = 0

Solve:

2cos(x)-1 = 0 or 2sin(x)+sqrt(3) = 0

cos(x) = 1/2 or sin(x) = -sqrt(3)/2

x = +/-pi/3 or x = -pi/3 or (4/3)pi

hope this helps!