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

The general solution for 2sin(θ)=sqrt(3)tan(θ) is θ=2pin,θ=pi+2pin,θ= pi/6+2pin,θ=(11pi)/6+2pin

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2sin(θ)=sqrt(3)tan(θ)

2 sin (θ) = 3 tan (θ)

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

Solution steps

2 sin (θ) = 3 tan (θ)

Subtract

3 tan (θ)

from both sides

2 sin (θ) - 3 tan (θ) = 0

Express with sin, cos

\frac{- sin ( θ ) 3 + 2 cos ( θ ) sin ( θ )}{cos ( θ )} = 0

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

- sin (θ) 3 + 2 cos (θ) sin (θ) = 0

Factor

- sin (θ) 3 + 2 cos (θ) sin (θ) : sin (θ) (- 3 + 2 cos (θ))

sin (θ) (- 3 + 2 cos (θ)) = 0

Solving each part separately

sin (θ) = 0 or - 3 + 2 cos (θ) = 0

sin (θ) = 0 : θ = 2 πn, θ = π + 2 πn

- 3 + 2 cos (θ) = 0 : θ = \frac{π}{6} + 2 πn, θ = \frac{11 π}{6} + 2 πn

Combine all the solutions

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

sin (x^{2} - 1) = 4

1 - cos (x) = \frac{2 π}{3}

sin (A) = \frac{1}{3}

sin (θ) = - 0.78

x, sin (2 x) + cos (x) = 0

What is the general solution for 2sin(θ)=sqrt(3)tan(θ) ?
The general solution for 2sin(θ)=sqrt(3)tan(θ) is θ=2pin,θ=pi+2pin,θ= pi/6+2pin,θ=(11pi)/6+2pin