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

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

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2sin(θ)=tan(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

sec (θ) = - 2.5

cos (4 x) = 0, 0 \leq x \leq 2 π

\frac{csc ( x )}{cot ( x )} = csc (x) sec (x)

sin (θ) = \frac{1}{3}

cos (θ) = 0.9

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