1. **Problem statement:** Given that $\tan \theta$ is undefined and $8\pi \leq \theta \leq 9\pi$, find $\sin \theta$, $\cot \theta$, $\cos \theta$, and also find $\csc \theta$ and $\sec \theta$. 2. **Understanding the problem:** The tangent function is undefined where $\cos \theta = 0$ because $\tan \theta = \frac{\sin \theta}{\cos \theta}$. So, we need to find $\theta$ in the interval $[8\pi, 9\pi]$ where $\cos \theta = 0$. 3. **Finding $\theta$ where $\cos \theta = 0$:** The cosine function is zero at $\theta = \frac{\pi}{2} + k\pi$ for any integer $k$. 4. **Find $k$ such that $8\pi \leq \frac{\pi}{2} + k\pi \leq 9\pi$:** $$8\pi \leq \frac{\pi}{2} + k\pi \leq 9\pi$$ Subtract $\frac{\pi}{2}$: $$8\pi - \frac{\pi}{2} \leq k\pi \leq 9\pi - \frac{\pi}{2}$$ $$\frac{16\pi}{2} - \frac{\pi}{2} \leq k\pi \leq \frac{18\pi}{2} - \frac{\pi}{2}$$ $$\frac{15\pi}{2} \leq k\pi \leq \frac{17\pi}{2}$$ Divide by $\pi$: $$\frac{15}{2} \leq k \leq \frac{17}{2}$$ $$7.5 \leq k \leq 8.5$$ Since $k$ is an integer, $k = 8$. 5. **Calculate $\theta$:** $$\theta = \frac{\pi}{2} + 8\pi = \frac{\pi}{2} + \frac{16\pi}{2} = \frac{17\pi}{2}$$ 6. **Find $\sin \theta$:** Since $\sin \theta = \sin \left( \frac{17\pi}{2} \right )$. Recall that $\sin(\theta)$ has period $2\pi$, so: $$\sin \left( \frac{17\pi}{2} \right ) = \sin \left( \frac{17\pi}{2} - 8\pi \right ) = \sin \left( \frac{17\pi}{2} - \frac{16\pi}{2} \right ) = \sin \left( \frac{\pi}{2} \right ) = 1$$ 7. **Find $\cot \theta$:** Since $\cot \theta = \frac{\cos \theta}{\sin \theta}$ and $\cos \theta = 0$ (because $\tan \theta$ undefined means $\cos \theta = 0$), $$\cot \theta = \frac{0}{\sin \theta} = 0$$ 8. **Find $\cos \theta$:** As established, $\cos \theta = 0$. 9. **Find $\csc \theta$:** $$\csc \theta = \frac{1}{\sin \theta} = \frac{1}{1} = 1$$ 10. **Find $\sec \theta$:** Since $\sec \theta = \frac{1}{\cos \theta}$ and $\cos \theta = 0$, $$\sec \theta = \text{undefined}$$ **Final answers:** - $\sin \theta = 1$ - $\cot \theta = 0$ - $\cos \theta = 0$ - $\csc \theta = 1$ - $\sec \theta$ is undefined