What is the general solution for -2cos^2(x)+3sin(x)+3=0 ?

The general solution for -2cos^2(x)+3sin(x)+3=0 is x=(7pi)/6+2pin,x=(11pi)/6+2pin,x=(3pi)/2+2pin

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-2cos^2(x)+3sin(x)+3=0

- 2 cos^{2} (x) + 3 sin (x) + 3 = 0

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

Solution steps

- 2 cos^{2} (x) + 3 sin (x) + 3 = 0

Rewrite using trig identities

1 + 2 sin^{2} (x) + 3 sin (x) = 0

Solve by substitution

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

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

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

Combine all the solutions

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

2 cos^{2} (x) = 3 \cdot cos (x)

4 (cos (x) + 1) cos (x) = 3

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

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

cos (2 a) = \frac{2}{3}, sin (a)

What is the general solution for -2cos^2(x)+3sin(x)+3=0 ?
The general solution for -2cos^2(x)+3sin(x)+3=0 is x=(7pi)/6+2pin,x=(11pi)/6+2pin,x=(3pi)/2+2pin