1. **Problem statement:** Given $\sin \theta = \frac{\sqrt{21}}{5}$ and $\frac{\pi}{2} \leq \theta \leq \pi$, find $\sin 2\theta$ and $\cos 2\theta$ without using a calculator. 2. **Recall formulas:** - Double angle formulas: $$\sin 2\theta = 2 \sin \theta \cos \theta$$ $$\cos 2\theta = \cos^2 \theta - \sin^2 \theta$$ 3. **Find $\cos \theta$:** Since $\sin^2 \theta + \cos^2 \theta = 1$, we have $$\cos^2 \theta = 1 - \sin^2 \theta = 1 - \left(\frac{\sqrt{21}}{5}\right)^2 = 1 - \frac{21}{25} = \frac{4}{25}$$ 4. **Determine the sign of $\cos \theta$:** Given $\frac{\pi}{2} \leq \theta \leq \pi$, $\theta$ is in the second quadrant where cosine is negative. So, $$\cos \theta = -\frac{2}{5}$$ 5. **Calculate $\sin 2\theta$:** $$\sin 2\theta = 2 \sin \theta \cos \theta = 2 \times \frac{\sqrt{21}}{5} \times \left(-\frac{2}{5}\right) = -\frac{4\sqrt{21}}{25}$$ 6. **Calculate $\cos 2\theta$:** $$\cos 2\theta = \cos^2 \theta - \sin^2 \theta = \left(-\frac{2}{5}\right)^2 - \left(\frac{\sqrt{21}}{5}\right)^2 = \frac{4}{25} - \frac{21}{25} = -\frac{17}{25}$$ **Final answers:** $$\sin 2\theta = -\frac{4\sqrt{21}}{25}, \quad \cos 2\theta = -\frac{17}{25}$$