Find the exact value of each of the remaining trigonometric functions of $\theta$.
\[
\cos \theta=\frac{2}{3}, \tan \theta< 0
\]

Step 1 ：Given that \(\cos \theta=\frac{2}{3}\), we can use the Pythagorean identity \(\sin^2 \theta + \cos^2 \theta = 1\) to find \(\sin \theta\). However, we need to consider the sign of \(\sin \theta\) based on the given condition \(\tan \theta<0\).

Step 2 ：Since \(\tan \theta = \frac{\sin \theta}{\cos \theta}\), and we know \(\cos \theta > 0\) (because \(\cos \theta=\frac{2}{3}\)), for \(\tan \theta\) to be negative, \(\sin \theta\) must be negative.

Step 3 ：So, we can calculate \(\sin \theta\) as \(-\sqrt{1 - \cos^2 \theta}\).

Step 4 ：After finding \(\sin \theta\), we can calculate \(\tan \theta = \frac{\sin \theta}{\cos \theta}\), \(\csc \theta = \frac{1}{\sin \theta}\), \(\sec \theta = \frac{1}{\cos \theta}\), and \(\cot \theta = \frac{1}{\tan \theta}\).

Step 5 ：The exact values of the remaining trigonometric functions of \(\theta\) are: \(\sin \theta = \boxed{-\frac{\sqrt{5}}{3}}\), \(\tan \theta = \boxed{-\sqrt{5}}\), \(\csc \theta = \boxed{-\frac{3}{\sqrt{5}}}\), \(\sec \theta = \boxed{\frac{3}{2}}\), \(\cot \theta = \boxed{-\frac{1}{\sqrt{5}}}\)