What is the general solution for sqrt(3)sin(θ)cot(θ)-2sin(θ)=sin(θ) ?

The general solution for sqrt(3)sin(θ)cot(θ)-2sin(θ)=sin(θ) is θ= pi/6+pin

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sqrt(3)sin(θ)cot(θ)-2sin(θ)=sin(θ)

3 sin (θ) cot (θ) - 2 sin (θ) = sin (θ)

θ = \frac{π}{6} + πn

Solution steps

3 sin (θ) cot (θ) - 2 sin (θ) = sin (θ)

Subtract

sin (θ)

from both sides

3 sin (θ) cot (θ) - 3 sin (θ) = 0

Factor

3 sin (θ) cot (θ) - 3 sin (θ) : sin (θ) (3 cot (θ) - 3)

sin (θ) (3 cot (θ) - 3) = 0

Solving each part separately

sin (θ) = 0 or 3 cot (θ) - 3 = 0

sin (θ) = 0 : θ = 2 πn, θ = π + 2 πn

3 cot (θ) - 3 = 0 : θ = \frac{π}{6} + πn

Combine all the solutions

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

Since the equation is undefined for:

2 πn, π + 2 πn

θ = \frac{π}{6} + πn

4 - 7 sin^{2} (θ) = 3 cos^{2} (θ)

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

7 sin^{2} (x) - 14 sin (x) + 2 = - 5

sin (x) = - 3

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

What is the general solution for sqrt(3)sin(θ)cot(θ)-2sin(θ)=sin(θ) ?
The general solution for sqrt(3)sin(θ)cot(θ)-2sin(θ)=sin(θ) is θ= pi/6+pin