\( \sin \theta=-\frac{7}{25}, \frac{3 \pi}{2}<\theta<2 \pi \) Find \( \sin \frac{\theta}{2} \)

To find \( \sin \frac{\theta}{2} \), we can use the half-angle identity: \[ \sin \frac{\theta}{2} = \pm \sqrt{\frac{1 - \cos \theta}{2}} \] First, we need to determine \( \cos \theta \). Since \( \sin^2 \theta + \cos^2 \theta = 1 \): \[ \cos^2 \theta = 1 - \sin^2 \theta = 1 - \left(-\frac{7}{25}\right)^2 = 1 - \frac{49}{625} = \frac{576}{625} \] Taking the square root gives: \[ \cos \theta = \pm \frac{24}{25} \] Since \( \frac{3\pi}{2} < \theta < 2\pi \), \(\cos \theta\) is positive in this range, so \( \cos \theta = \frac{24}{25} \). Now we can use this to find \( \sin \frac{\theta}{2} \): \[ \sin \frac{\theta}{2} = \sqrt{\frac{1 - \frac{24}{25}}{2}} = \sqrt{\frac{\frac{1}{25}}{2}} = \sqrt{\frac{1}{50}} = \frac{1}{5\sqrt{2}} = \frac{\sqrt{2}}{10} \] Since \(\theta\) is between \( \frac{3\pi}{2} \) and \( 2\pi\), \(\frac{\theta}{2}\) is between \( \frac{3\pi}{4} \) and \( \pi\), meaning \(\sin \frac{\theta}{2}\) is positive. Therefore, the final answer is: \[ \sin \frac{\theta}{2} = \frac{\sqrt{2}}{10} \]