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

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

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sin^2(θ)+2-cos^2(θ)=3sin(θ)

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

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

Solution steps

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

Subtract

3 sin (θ)

from both sides

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

Rewrite using trig identities

1 + 2 sin^{2} (θ) - 3 sin (θ) = 0

Solve by substitution

sin (θ) = 1, sin (θ) = \frac{1}{2}

sin (θ) = 1 : θ = \frac{π}{2} + 2 πn

sin (θ) = \frac{1}{2} : θ = \frac{π}{6} + 2 πn, θ = \frac{5 π}{6} + 2 πn

Combine all the solutions

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

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

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

tan^{2} (x) + 2 tan (x) + 1 = 0

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

4 sin (x) + 5 = 4 + sin (x)

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