6 ANS:
20 If $\cos A=\frac{\sqrt{5}}{3}$ and $\tan A< 0$, what is the value of $\sin A$ ?
(1) $\frac{2}{3}$
(3) $-\frac{2}{3}$
(2) $-\frac{\sqrt{5}}{3}$
(4) $\frac{3}{\sqrt{5}}$

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 \(\boxed{-\frac{2}{3}}\)