Step 1 :Given that \(\cos A = \frac{\sqrt{5}}{3}\) and \(\tan A < 0\), we need to find the value of \(\sin A\).
Step 2 :Using the Pythagorean identity \(\sin^2 A + \cos^2 A = 1\), we can find \(\sin A\).
Step 3 :\(\sin^2 A = 1 - \cos^2 A = 1 - \left(\frac{\sqrt{5}}{3}\right)^2 = \frac{4}{9}\)
Step 4 :Since \(\sin^2 A = \frac{4}{9}\), we have two possible values for \(\sin A\): \(\pm\frac{2}{3}\).
Step 5 :However, we are given that \(\tan A < 0\). Since \(\cos A > 0\) and \(\tan A = \frac{\sin A}{\cos A}\), this means that \(\sin A\) must be negative.
Step 6 :Thus, the correct value of \(\sin A\) is \(\boxed{-\frac{2}{3}}\)