Find all points of intersection of the given curves. (Assume 0 ≤ θ < 2π and r ≥ 0. Order your answers from smallest to largest θ. If an intersection occurs at the pole, enter POLE in the first answer blank.) r = sin(θ), r = sin(2θ)

Answer:[tex]\theta=0(\:pole),\frac{\pi}{3},\frac{5\pi}{3}[/tex]Step-by-step explanation:The given system of polar equations are:[tex]r=\sin \theta[/tex][tex]r=\sin 2\theta[/tex]We equate both equations to get:[tex]\sin 2\theta=\sin \theta[/tex]We rewrite to get:[tex]\sin 2\theta-\sin \theta=0[/tex]Apply the double angle property to get:[tex]2\sin \theta \cos \theta-\sin \theta=0[/tex]Factor now to get:[tex]\sin \theta( 2\cos \theta-1)=0[/tex][tex]\implies \sin \theta=0\:or\:\cos \theta=0.5[/tex]Hence [tex]\theta=0,\frac{\pi}{3},\frac{5\pi}{3}[/tex]