What is the general solution for solvefor θ,2sin(θ)=tan(θ) ?

The general solution for solvefor θ,2sin(θ)=tan(θ) is θ=2pin,θ=pi+2pin,θ= pi/3+2pin,θ=(5pi)/3+2pin

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solvefor θ,2sin(θ)=tan(θ)

solve for

θ, 2 sin (θ) = tan (θ)

θ = 2 πn, θ = π + 2 πn, θ = \frac{π}{3} + 2 πn, θ = \frac{5 π}{3} + 2 πn

Solution steps

2 sin (θ) = tan (θ)

Subtract

tan (θ)

from both sides

2 sin (θ) - tan (θ) = 0

Express with sin, cos

\frac{- sin ( θ ) + 2 cos ( θ ) sin ( θ )}{cos ( θ )} = 0

\frac{f ( x )}{g ( x )} = 0 \Rightarrow f (x) = 0

- sin (θ) + 2 cos (θ) sin (θ) = 0

Factor

- sin (θ) + 2 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{π}{3} + 2 πn, θ = \frac{5 π}{3} + 2 πn

Combine all the solutions

θ = 2 πn, θ = π + 2 πn, θ = \frac{π}{3} + 2 πn, θ = \frac{5 π}{3} + 2 πn

5 cot^{2} (x) + 8 cot (x) + 3 = 0

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

2 tan^{2} (2 x) - tan (2 x) = 3

sin (x) = \frac{15}{31}

4 cot (θ) - 1 = 0

What is the general solution for solvefor θ,2sin(θ)=tan(θ) ?
The general solution for solvefor θ,2sin(θ)=tan(θ) is θ=2pin,θ=pi+2pin,θ= pi/3+2pin,θ=(5pi)/3+2pin