What is the general solution for 2sin^2(x)-sin(x)=1 ?

The general solution for 2sin^2(x)-sin(x)=1 is x= pi/2+2pin,x=(7pi)/6+2pin,x=(11pi)/6+2pin

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2sin^2(x)-sin(x)=1

2 sin^{2} (x) - sin (x) = 1

x = \frac{π}{2} + 2 πn, x = \frac{7 π}{6} + 2 πn, x = \frac{11 π}{6} + 2 πn

Solution steps

2 sin^{2} (x) - sin (x) = 1

Solve by substitution

sin (x) = 1, sin (x) = - \frac{1}{2}

sin (x) = 1 : x = \frac{π}{2} + 2 πn

sin (x) = - \frac{1}{2} : x = \frac{7 π}{6} + 2 πn, x = \frac{11 π}{6} + 2 πn

Combine all the solutions

x = \frac{π}{2} + 2 πn, x = \frac{7 π}{6} + 2 πn, x = \frac{11 π}{6} + 2 πn

cos^{2} (x) - sin^{2} (x) = \frac{1}{2}

2 cos^{2} (θ) - cos (θ) = 0

3 tan^{2} (θ) - 1 = 0

sin (2 θ - \frac{π}{2}) = - 1

sin (\frac{x}{2}) + cos (x) = 0

What is the general solution for 2sin^2(x)-sin(x)=1 ?
The general solution for 2sin^2(x)-sin(x)=1 is x= pi/2+2pin,x=(7pi)/6+2pin,x=(11pi)/6+2pin