Let $g(\theta)=1-4\sin\theta+6\cos3\theta$. As in the problem, at the maxima $\theta=0,\frac{2\pi}3,\frac{4\pi}3,2\pi$ of $\cos3\theta$ we have $g=7-4\sin\theta\ge3>0$, and at the minima $\theta=\frac\pi3,\pi,\frac{5\pi}3$ we have $g=-5-4\sin\theta\le-1<0$. So each of the $6$ intervals between consecutive extreme points contains a root.
Where $|\sin3\theta|>\frac29$, $|18\sin3\theta|>4\ge|4\cos\theta|$, so $g'=-4\cos\theta-18\sin3\theta$ has a fixed sign and $g$ is monotone. Where $|\sin3\theta|\le\frac29$, $|\cos3\theta|\ge\frac{\sqrt{77}}9>0.97$, so $g>6(0.97)+1-4>0$ near a maximum and $g<-6(0.97)+1+4<0$ near a minimum: no roots there.
So each interval contains exactly one root, and there are $6$ roots.
