What is the general solution for 2sin(θ)-1=csc(θ) ?

The general solution for 2sin(θ)-1=csc(θ) is θ=(7pi)/6+2pin,θ=(11pi)/6+2pin,θ= pi/2+2pin

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2sin(θ)-1=csc(θ)

2 sin (θ) - 1 = csc (θ)

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

Solution steps

2 sin (θ) - 1 = csc (θ)

Subtract

csc (θ)

from both sides

2 sin (θ) - 1 - csc (θ) = 0

Rewrite using trig identities

- 1 - csc (θ) + \frac{2}{csc ( θ )} = 0

Solve by substitution

csc (θ) = - 2, csc (θ) = 1

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

csc (θ) = 1 : θ = \frac{π}{2} + 2 πn

Combine all the solutions

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

sin^{2} (x) + 3 sin (x) - 4 = 0

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

2 tan^{2} (x) + 3 sec (x) = 0

tan (x) = - \frac{3}{4}

tan (x) = - \frac{3}{2}

What is the general solution for 2sin(θ)-1=csc(θ) ?
The general solution for 2sin(θ)-1=csc(θ) is θ=(7pi)/6+2pin,θ=(11pi)/6+2pin,θ= pi/2+2pin