Use trigonometric identities to solve the trigonometric equation $\sin (2\theta )+\cos \theta =0$ exactly for the domain $0\le \theta \le 2\pi$.

Step 1 ：Given the trigonometric equation $\sin (2\theta )+\cos \theta =0$.

Step 2 ：Using the trigonometric identity $\sin (2\theta )=2\sin \theta \cos \theta$, the equation can be rewritten as $2\sin \theta \cos \theta +\cos \theta =0$.

Step 3 ：Factoring out $\cos \theta$, we get $\cos \theta (2\sin \theta +1)=0$.

Step 4 ：This gives us two equations to solve: $\cos \theta =0$ and $2\sin \theta +1=0$.

Step 5 ：The solutions to the equation $\cos \theta =0$ are $\theta =\frac{\pi }{2}$ and $\theta =\frac{3\pi }{2}$.

Step 6 ：The solution to the equation $2\sin \theta +1=0$ is $\theta =\frac{7\pi }{4}$.

Step 7 ：These are the solutions in the domain $0\le \theta \le 2\pi$.

Step 8 ：Final Answer: The solutions to the equation $\sin (2\theta )+\cos \theta =0$ in the domain $0\le \theta \le 2\pi$ are $\theta =\frac{\pi }{2}$, $\theta =\frac{3\pi }{2}$, and $\theta =\frac{7\pi }{4}$.

Step 9 ：So, the final answer is $\boxed{{\displaystyle \theta =\frac{\pi }{2},\theta =\frac{3\pi }{2},\theta =\frac{7\pi }{4}}}$.