What is the general solution for -2sin^2(x)=3sin(x)+1 ?

The general solution for -2sin^2(x)=3sin(x)+1 is x=(3pi)/2+2pin,x=(7pi)/6+2pin,x=(11pi)/6+2pin

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-2sin^2(x)=3sin(x)+1

- 2 sin^{2} (x) = 3 sin (x) + 1

x = \frac{3 π}{2} + 2 πn, x = \frac{7 π}{6} + 2 πn, x = \frac{11 π}{6} + 2 πn

Solution steps

- 2 sin^{2} (x) = 3 sin (x) + 1

Solve by substitution

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

sin (x) = - 1 : x = \frac{3 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{3 π}{2} + 2 πn, x = \frac{7 π}{6} + 2 πn, x = \frac{11 π}{6} + 2 πn

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

2 csc^{2} (x) - 3 csc (x) - 2 = 0

csc (x) = \frac{2}{3}

8 - 7 cos (x) = 6 sin^{2} (x)

2 = 4 cos^{2} (x) + 1

What is the general solution for -2sin^2(x)=3sin(x)+1 ?
The general solution for -2sin^2(x)=3sin(x)+1 is x=(3pi)/2+2pin,x=(7pi)/6+2pin,x=(11pi)/6+2pin