We can firstly factorise our original equation to make it become:
sintheta(1+2costheta)=0
Therefore, if we set the two parts to zero separately that will make the equation work. For example, if sintheta=0, then 0*(whatever the other part becomes) will indeed =0.
So, separately we will make the two parts =0. Starting with the left part, if sintheta=0, theta=0^o, 180^o,360^o... For the remainder of this question I will consider only the domain 0^o<=theta<=360^o.
Then, we make the other part =0. This means that 1+2costheta=0, by rearranging this becomes costheta=-1/2. By solving with a calculator, our principal value is 120^o. 240^o is also a solution since if we consider how costheta reflects then we can obtain the above solution by doing 360-120.
Our final solutions for theta are 0^o, 120^o, 180^o, 240^o and 360^o, within the restricted domain 0^o<=theta<=360^o. You can test all of these by substituting them into the original equation and they will indeed produce zero.
