Answer
$\theta=n\pi$, $\theta=\frac{\pi}{3}\pm 2n\pi$, $n\in Z$
Work Step by Step
$R'(\theta)=-\sin\theta+2\cos\theta\sin\theta$
$=\sin\theta(2\cos \theta -1)$
$R'(\theta)$ exists for all $\theta$.
$R'(\theta)= 0\implies $$\sin\theta=0$ or $2\cos\theta-1=0$
$\implies \theta= n\pi$ or $\cos\theta=\frac{1}{2}$
$\implies \theta=n\pi$ or $\theta =\frac{\pi}{3}\pm 2n\pi$, where $n$ is any integer.
That is, the critical points of the function are all $\theta=n\pi $ or $\theta=\frac{\pi}{3}\pm 2n\pi$ where $n$ is any integer.
