1. **State the problem:** Given $\cos \theta = -\frac{2}{3}$ and $\pi < \theta < \frac{3\pi}{2}$, find $\sin \frac{\theta}{2}$, $\cos \frac{\theta}{2}$, and $\tan \frac{\theta}{2}$. 2. **Identify the quadrant of $\theta$:** Since $\pi < \theta < \frac{3\pi}{2}$, $\theta$ is in the third quadrant where sine is negative and cosine is negative. 3. **Find $\sin \theta$ using Pythagorean identity:** $$\sin^2 \theta = 1 - \cos^2 \theta = 1 - \left(-\frac{2}{3}\right)^2 = 1 - \frac{4}{9} = \frac{5}{9}.$$ Since $\theta$ is in the third quadrant, $\sin \theta < 0$, so $$\sin \theta = -\sqrt{\frac{5}{9}} = -\frac{\sqrt{5}}{3}.$$ 4. **Use half-angle formulas:** $$\sin \frac{\theta}{2} = \pm \sqrt{\frac{1 - \cos \theta}{2}}, \quad \cos \frac{\theta}{2} = \pm \sqrt{\frac{1 + \cos \theta}{2}}, \quad \tan \frac{\theta}{2} = \frac{\sin \theta}{1 + \cos \theta}.$$ 5. **Determine the quadrant of $\frac{\theta}{2}$:** Since $\pi < \theta < \frac{3\pi}{2}$, dividing by 2 gives $$\frac{\pi}{2} < \frac{\theta}{2} < \frac{3\pi}{4}.$$ This places $\frac{\theta}{2}$ in the second quadrant where sine is positive and cosine is negative. 6. **Calculate $\sin \frac{\theta}{2}$:** $$\sin \frac{\theta}{2} = +\sqrt{\frac{1 - \left(-\frac{2}{3}\right)}{2}} = \sqrt{\frac{1 + \frac{2}{3}}{2}} = \sqrt{\frac{\frac{3}{3} + \frac{2}{3}}{2}} = \sqrt{\frac{\frac{5}{3}}{2}} = \sqrt{\frac{5}{6}} = \frac{\sqrt{30}}{6}.$$ 7. **Calculate $\cos \frac{\theta}{2}$:** $$\cos \frac{\theta}{2} = -\sqrt{\frac{1 + \left(-\frac{2}{3}\right)}{2}} = -\sqrt{\frac{1 - \frac{2}{3}}{2}} = -\sqrt{\frac{\frac{3}{3} - \frac{2}{3}}{2}} = -\sqrt{\frac{\frac{1}{3}}{2}} = -\sqrt{\frac{1}{6}} = -\frac{\sqrt{6}}{6}.$$ 8. **Calculate $\tan \frac{\theta}{2}$:** Using the formula $$\tan \frac{\theta}{2} = \frac{\sin \theta}{1 + \cos \theta} = \frac{-\frac{\sqrt{5}}{3}}{1 - \frac{2}{3}} = \frac{-\frac{\sqrt{5}}{3}}{\frac{1}{3}} = -\sqrt{5}.$$ **Final answers:** $$\sin \frac{\theta}{2} = \frac{\sqrt{30}}{6}, \quad \cos \frac{\theta}{2} = -\frac{\sqrt{6}}{6}, \quad \tan \frac{\theta}{2} = -\sqrt{5}.$$