What is the general solution for tan(θ)=2sin(θ),0<= θ<= 2pi ?

The general solution for tan(θ)=2sin(θ),0<= θ<= 2pi is θ=0,θ=pi,θ=2pi,θ= pi/3 ,θ=(5pi)/3

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tan(θ)=2sin(θ),0<= θ<= 2pi

tan (θ) = 2 sin (θ), 0 \leq θ \leq 2 π

θ = 0, θ = π, θ = 2 π, θ = \frac{π}{3}, θ = \frac{5 π}{3}

Solution steps

tan (θ) = 2 sin (θ), 0 \leq θ \leq 2 π

Subtract

2 sin (θ)

from both sides

tan (θ) - 2 sin (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, 0 \leq θ \leq 2 π : θ = 0, θ = π, θ = 2 π

2 cos (θ) - 1 = 0, 0 \leq θ \leq 2 π : θ = \frac{π}{3}, θ = \frac{5 π}{3}

Combine all the solutions

θ = 0, θ = π, θ = 2 π, θ = \frac{π}{3}, θ = \frac{5 π}{3}

2 sin (x) tan (x) - tan (x) = 0

sin (x) = sin (3 x)

7 tan (θ) + 5 = 0

sin (x + 4 5^{\circ}) = \frac{3}{2}

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

What is the general solution for tan(θ)=2sin(θ),0<= θ<= 2pi ?
The general solution for tan(θ)=2sin(θ),0<= θ<= 2pi is θ=0,θ=pi,θ=2pi,θ= pi/3 ,θ=(5pi)/3