QUESTION 9 - 1 POINT
If $\sin (\theta )=\frac{7}{25}$, and $\theta$ is in Quadrant II, then what is $\sin \left(\frac{\theta }{2}\right)$ ? Type an exact answer, using radicals as needed. Simplify your answer completely and rationalize the denominator.

Provide your answer below:

Step 1 ：${\sin }^2(\theta )+{\cos }^2(\theta )=1$

Step 2 ：${\cos }^2(\theta )=1-{\sin }^2(\theta )$

Step 3 ：${\cos }^2(\theta )=1-{\left(\frac{7}{25}\right)}^2$

Step 4 ：${\cos }^2(\theta )=1-\frac{49}{625}$

Step 5 ：${\cos }^2(\theta )=\frac{625}{625}-\frac{49}{625}$

Step 6 ：${\cos }^2(\theta )=\frac{576}{625}$

Step 7 ：$\cos (\theta )=\pm \frac{24}{25}$

Step 8 ：$\cos (\theta )=-\frac{24}{25}$ (\text{since } \theta \text{ is in Quadrant II})

Step 9 ：$\sin \left(\frac{\theta }{2}\right)=\pm \sqrt{\frac{1-\cos (\theta )}{2}}$

Step 10 ：$\sin \left(\frac{\theta }{2}\right)=\pm \sqrt{\frac{1-(-\frac{24}{25})}{2}}$

Step 11 ：$\sin \left(\frac{\theta }{2}\right)=\pm \sqrt{\frac{1+\frac{24}{25}}{2}}$

Step 12 ：$\sin \left(\frac{\theta }{2}\right)=\pm \sqrt{\frac{25}{25}+\frac{24}{25}}$

Step 13 ：$\sin \left(\frac{\theta }{2}\right)=\pm \sqrt{\frac{49}{25}}$

Step 14 ：$\sin \left(\frac{\theta }{2}\right)=\pm \sqrt{\frac{49}{25}\cdot \frac{2}{2}}$

Step 15 ：$\sin \left(\frac{\theta }{2}\right)=\pm \sqrt{\frac{98}{50}}$

Step 16 ：$\sin \left(\frac{\theta }{2}\right)=\pm \sqrt{\frac{49}{25}}$

Step 17 ：$\sin \left(\frac{\theta }{2}\right)=\pm \frac{7}{5}\sqrt{\frac{1}{2}}$

Step 18 ：$\sin \left(\frac{\theta }{2}\right)=\pm \frac{7}{5}\cdot \frac{\sqrt{2}}{2}$

Step 19 ：$\sin \left(\frac{\theta }{2}\right)=\pm \frac{7\sqrt{2}}{10}$

Step 20 ：$\sin \left(\frac{\theta }{2}\right)=\frac{7\sqrt{2}}{10}$ (\text{since } \frac{\theta}{2} \text{ will be in Quadrant I})

Step 21 ：$\boxed{{\displaystyle \sin \left(\frac{\theta }{2}\right)=\frac{7\sqrt{2}}{10}}}$