What is the general solution for 2sin(θ)cos(θ)+sin(θ)=0 ?

The general solution for 2sin(θ)cos(θ)+sin(θ)=0 is θ=2pin,θ=pi+2pin,θ=(2pi)/3+2pin,θ=(4pi)/3+2pin

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2sin(θ)cos(θ)+sin(θ)=0

2 sin (θ) cos (θ) + sin (θ) = 0

θ = 2 πn, θ = π + 2 πn, θ = \frac{2 π}{3} + 2 πn, θ = \frac{4 π}{3} + 2 πn

Solution steps

2 sin (θ) cos (θ) + sin (θ) = 0

Factor

2 sin (θ) cos (θ) + sin (θ) : sin (θ) (2 cos (θ) + 1)

sin (θ) (2 cos (θ) + 1) = 0

Solving each part separately

sin (θ) = 0 or 2 cos (θ) + 1 = 0

sin (θ) = 0 : θ = 2 πn, θ = π + 2 πn

2 cos (θ) + 1 = 0 : θ = \frac{2 π}{3} + 2 πn, θ = \frac{4 π}{3} + 2 πn

Combine all the solutions

θ = 2 πn, θ = π + 2 πn, θ = \frac{2 π}{3} + 2 πn, θ = \frac{4 π}{3} + 2 πn

csc (3 x) - 8 = 0

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

sin (x) \cdot cos (x) = \frac{1}{2}

cos (x) sin (x) - cos (x) = 0

sin^{2} (θ) = sin (θ)

What is the general solution for 2sin(θ)cos(θ)+sin(θ)=0 ?
The general solution for 2sin(θ)cos(θ)+sin(θ)=0 is θ=2pin,θ=pi+2pin,θ=(2pi)/3+2pin,θ=(4pi)/3+2pin