tan^3(θ)=3tan(θ)

solve for all solutions..
please provide steps so I can follow

tan^2 T = 3 and tan T = 0

so
tan T = sqrt 3
tan T = -sqrt 3
that gives solutions in every quadrant
plus
T = 0 when T = 180 deg or pi radians

To solve the equation tan^3(θ) = 3tan(θ) for all solutions, follow these steps:

Step 1: Rewrite the equation using the identity tan^3(θ) = tan(θ)*tan^2(θ). The equation becomes tan(θ)*tan^2(θ) = 3tan(θ).

Step 2: Divide both sides of the equation by tan(θ) to get tan^2(θ) = 3.

Step 3: Take the square root of both sides of the equation to eliminate the square on the left side. You will have two cases to consider: tan(θ) = sqrt(3) and tan(θ) = -sqrt(3).

Step 4: Solve for θ by using the inverse tangent function (tan^-1) on both sides of each equation.

For the first case, tan(θ) = sqrt(3):
θ = tan^-1(sqrt(3)) + kπ, where k is an integer.

For the second case, tan(θ) = -sqrt(3):
θ = tan^-1(-sqrt(3)) + kπ, where k is an integer.

By following these steps, you can solve the equation tan^3(θ) = 3tan(θ) for all solutions.