What is the general solution for csc^2(θ)+3csc(θ)+2=0 ?

The general solution for csc^2(θ)+3csc(θ)+2=0 is θ=(3pi)/2+2pin,θ=(7pi)/6+2pin,θ=(11pi)/6+2pin

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csc^2(θ)+3csc(θ)+2=0

csc^{2} (θ) + 3 csc (θ) + 2 = 0

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

Solution steps

csc^{2} (θ) + 3 csc (θ) + 2 = 0

Solve by substitution

csc (θ) = - 1, csc (θ) = - 2

csc (θ) = - 1 : θ = \frac{3 π}{2} + 2 πn

csc (θ) = - 2 : θ = \frac{7 π}{6} + 2 πn, θ = \frac{11 π}{6} + 2 πn

Combine all the solutions

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

cos^{2} (x) = 3 - 5 sin (x)

cos^{2} (θ) - sin^{2} (θ) = - 1

8 sin (\frac{x}{2}) + 8 cos (x) = 0

tan (\frac{θ}{2} + \frac{π}{6}) = 1

sec (x) = - 2, 0 < x < 2 π

What is the general solution for csc^2(θ)+3csc(θ)+2=0 ?
The general solution for csc^2(θ)+3csc(θ)+2=0 is θ=(3pi)/2+2pin,θ=(7pi)/6+2pin,θ=(11pi)/6+2pin