Given $\cos \theta=\frac{\sqrt{3}}{3}$ and $\sin \theta< 0$ What is the value of $\sin \theta$ ?

Step 1 ：The given value of \(\cos \theta\) is positive and \(\sin \theta\) is negative. This implies that the angle \(\theta\) is in the fourth quadrant. In the fourth quadrant, the value of \(\sin \theta\) is negative and the value of \(\cos \theta\) is positive.

Step 2 ：We know that \(\sin^2 \theta + \cos^2 \theta = 1\). We can use this identity to find the value of \(\sin \theta\).

Step 3 ：Given \(\cos \theta = 0.5773502691896257\), we can substitute this into the identity to find \(\sin \theta\).

Step 4 ：Solving for \(\sin \theta\), we get \(\sin \theta = -0.816496580927726\).

Step 5 ：Final Answer: The value of \(\sin \theta\) is \(\boxed{-0.816496580927726}\).