Use a Double - or Half-Angle Formula to solve the equation in the interval [0, 2π). (Enter your answers as a comma-separated list. Round your answers to three decimal places where appropriate.)

sin 2θ + sin θ = 0

Sin (2θ) + sin (θ) = 0

use double angle formula: sin (2 θ) = 2sin (θ) cos (θ).

=>

2sin (θ) cos (θ) + sin (θ) = 0

factor out sin (θ)

sin (θ) (2cos (θ) + 1) = 0

by the zero product property,

sin (θ) = 0 ... (a) or

(2cos (θ) + 1) = 0 ... (b)

Solution to (a) : θ=k (π)

solution to (b) : θ = (2k+1) (π) + / - (π) / 3

for k=integer

For [0,2 π), this translates to:

{0, 2 π/3,π,4π/3}