find all values of theta satisfying the equation

Question

2 tan theta sin theta + square root of 3 tan theta = 0

Answer 1

2 tanθ sinθ + sqrt(3) tanθ = 0

Factor: tanθ (2sin θ + sqrt(3)) = 0

tanθ = 0 ⇒ θ = 0, π in (0, 2π)

2sinθ = -sqrt(3) ⇒ θ = (4π)/3, (5π)/3 in (0, 2π)

Since the interval is not specified, the answer needs to be modified to include all allowable angles:

θ = nπ, θ = (4π)/3 + 2nπ, θ = (5π)/3 + 2nπ, where n = the set of integers: n = ...-3, -2, -1, 0, 1, 2, 3, ...