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

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 \(0 \leq \theta \leq 2 \pi\).

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

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