Given that $\sin \theta=\frac{35}{37}$ and $\cos \theta< 0$, determine the values of the sine and $\operatorname{cosine}$ functions for $2 \theta$.
\[
\sin 2 \theta=\square \text { (Type an integer or a simplified fraction.) }
\]

Step 1 ：We are given that \(\sin\theta=\frac{35}{37}\) and \(\cos\theta<0\).

Step 2 ：We can find the value of \(\cos\theta\) using the Pythagorean identity \(\sin^2\theta + \cos^2\theta = 1\). Since \(\cos\theta\) is negative, we will take the negative square root.

Step 3 ：After finding the value of \(\cos\theta\), we can substitute the values of \(\sin\theta\) and \(\cos\theta\) into the formula for \(\sin 2\theta\) to find the answer.

Step 4 ：\(\sin\theta = 0.9459459459459459\)

Step 5 ：\(\cos\theta = -0.32432432432432434\)

Step 6 ：\(\sin 2\theta = -0.6135865595325055\)

Step 7 ：Final Answer: \(\sin 2 \theta = \boxed{-\frac{613}{1000}}\)