Find $\sin \theta$ if $\cos \theta=\frac{2}{3}$ and $\theta$ is in quadrant IV

Step 1 ：We are given that \(\cos \theta = \frac{2}{3}\) and \(\theta\) is in quadrant IV.

Step 2 ：In the fourth quadrant, the sine function is negative.

Step 3 ：We can use the Pythagorean identity \(\sin^2 \theta + \cos^2 \theta = 1\) to find the value of \(\sin \theta\).

Step 4 ：Substituting \(\cos \theta = \frac{2}{3}\) into the identity, we get \(\sin^2 \theta = 1 - \cos^2 \theta\).

Step 5 ：Solving for \(\sin \theta\), we get \(\sin \theta = -\sqrt{1 - \cos^2 \theta}\).

Step 6 ：Substituting \(\cos \theta = \frac{2}{3}\) into the equation, we get \(\sin \theta = -\sqrt{1 - \left(\frac{2}{3}\right)^2}\).

Step 7 ：Solving the equation, we get \(\sin \theta = -0.7453559924999299\).

Step 8 ：Final Answer: \(\boxed{-0.7453559924999299}\)