To solve this equation, we first need to rewrite it in a more manageable form.
Given equation: 2(sin^2x + sinx) = sqrt(3) + sqrt(3)sinx
We know that sin^2x = 1 - cos^2x, so we can rewrite sin^2x in terms of cosx.
Rewritten equation: 2(1 - cos^2x + sinx) = sqrt(3) + sqrt(3)sinx
Expand and simplify the equation:
2 - 2cos^2x + 2sinx = sqrt(3) + sqrt(3)sinx
Rearrange the terms:
2sinx + sqrt(3)sinx + 2 = 2cos^2x + sqrt(3)
Combine like terms:
3sinx + 2 = 2cos^2x + sqrt(3)
Since sinx = cos(90° - x), we can substitute cos^2x with sinx:
3sinx + 2 = 2sin^2x + sqrt(3)
Rearrange the equation:
0 = 2sin^2x - 3sinx + sqrt(3) - 2
Now we have a quadratic equation in terms of sinx:
2sin^2x - 3sinx + sqrt(3) - 2 = 0
To solve this equation, we can use the quadratic formula:
sinx = [3 ± sqrt(9 - 4(2)(sqrt(3) - 2))]/(4)
sinx = [3 ± sqrt(9 - 8sqrt(3) + 8)]/(4)
sinx = [3 ± sqrt(17 - 8sqrt(3))]/4
Therefore, the solutions to the equation are sinx = [3 + sqrt(17 - 8sqrt(3))]/4 and sinx = [3 - sqrt(17 - 8sqrt(3))]/4.
To solve this equation, we first need to rewrite it in a more manageable form.
Given equation: 2(sin^2x + sinx) = sqrt(3) + sqrt(3)sinx
We know that sin^2x = 1 - cos^2x, so we can rewrite sin^2x in terms of cosx.
Rewritten equation: 2(1 - cos^2x + sinx) = sqrt(3) + sqrt(3)sinx
Expand and simplify the equation:
2 - 2cos^2x + 2sinx = sqrt(3) + sqrt(3)sinx
Rearrange the terms:
2sinx + sqrt(3)sinx + 2 = 2cos^2x + sqrt(3)
Combine like terms:
3sinx + 2 = 2cos^2x + sqrt(3)
Since sinx = cos(90° - x), we can substitute cos^2x with sinx:
3sinx + 2 = 2sin^2x + sqrt(3)
Rearrange the equation:
0 = 2sin^2x - 3sinx + sqrt(3) - 2
Now we have a quadratic equation in terms of sinx:
2sin^2x - 3sinx + sqrt(3) - 2 = 0
To solve this equation, we can use the quadratic formula:
sinx = [3 ± sqrt(9 - 4(2)(sqrt(3) - 2))]/(4)
sinx = [3 ± sqrt(9 - 8sqrt(3) + 8)]/(4)
sinx = [3 ± sqrt(17 - 8sqrt(3))]/4
Therefore, the solutions to the equation are sinx = [3 + sqrt(17 - 8sqrt(3))]/4 and sinx = [3 - sqrt(17 - 8sqrt(3))]/4.