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

The general solution for sin^2(x)=-2+3sin(x)+cos^2(x) is x= pi/2+2pin,x= pi/6+2pin,x=(5pi)/6+2pin

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

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

x = \frac{π}{2} + 2 πn, x = \frac{π}{6} + 2 πn, x = \frac{5 π}{6} + 2 πn

Solution steps

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

Subtract

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

from both sides

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

Rewrite using trig identities

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

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{π}{6} + 2 πn, x = \frac{5 π}{6} + 2 πn

Combine all the solutions

x = \frac{π}{2} + 2 πn, x = \frac{π}{6} + 2 πn, x = \frac{5 π}{6} + 2 πn

sin (x) - 3 cos (x) = - 1

sin (θ) = \frac{- 1}{17}

tan (x) + cot (x) = 3, = tan (3 x) + cot (3 x)

x, cos^{2} (x) = sin^{2} (x)

3 cot (x) - 1 = cot^{2} (x)

What is the general solution for sin^2(x)=-2+3sin(x)+cos^2(x) ?
The general solution for sin^2(x)=-2+3sin(x)+cos^2(x) is x= pi/2+2pin,x= pi/6+2pin,x=(5pi)/6+2pin