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

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

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2sin(θ)+3=-csc(θ)

2 sin (θ) + 3 = - csc (θ)

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

Solution steps

2 sin (θ) + 3 = - csc (θ)

Subtract

- csc (θ)

from both sides

2 sin (θ) + 3 + csc (θ) = 0

Rewrite using trig identities

3 + csc (θ) + \frac{2}{csc ( θ )} = 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

2 sin (\frac{2 x}{3}) - 2 = 0

tan (θ) = - \frac{1}{4}

\frac{1}{cos ( x ) + 1} + \frac{1}{cos ( x ) - 1} = 2 csc (x) cot (x)

3 - sin (θ) = cos (2 θ)

tan (θ) = \frac{- 1}{1}

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