Find all solutions for the equation 2sin^2(x)=-sin(x) from 0 ≤ x ≤ 2π

Answer 1

2sin^2(x) =-sin(x)

divide both sides by sin(x)

2sin(x) =-1

divide both sides by 2

sin(x) =-1/2

x = sin^-1(1/2)

x = 210 or 330 degrees = 7pi/6 or 11pi/6 radians

but also

sin(x) = 0

x = sin^-1(0) = 0 or 180 degrees or pi radians or 0 radians

x = 0, 180, 210, or 330 degrees

or x = 0, pi, 7pi/6 or 11pi/6 radians

or another method, let sin(x) = y

then

2y^2 =-y

2y^2 +y = 0

y(2y+1) = 0

y= 0 and y =-1/2

sin(x) =0 and sin(x) = -1/2

x = 0, pi, 7pi/6 and 11pi/6