3tan^3(theta)=tan(theta) - Find all complex solutions

Question

3tan³(theta)=tan(theta) - Find all complex solutions

Answer 1

let x=θ

3tan^3x=tanx

3tan^3x-tanx=0

factor

tanx(3tan^2x-1)=0

tanx=0 and 3tan^2x-1=0

tanx=0 when x=0°, 180°, and 360°

3tan^2x-1=0

3tan^2x=1

tan^2x=1/3

tanx=±√(1/3)

tanx=±√3/3

tan=sin/cos

look on the unit circle

tanx=√3/3 when x=30° and 210°

tanx=-√3/3 when x=150° and 330°

all solutions: x=0°, 30°, 150°, 180°, 210°, 330°, and 360°

Answer 2

3 tan³ θ = tan θ

3 tan² θ = 1 Assuming tan θ ≠ 0

tan² θ = 1 / 3

tan θ = ±√3 / 3

θ = π / 6

θ = 11π / 6