Use the information given about the angle \( \theta, \cot \theta=-8, \sec \theta<0,0 \leq \theta<2 \pi \), to find the exact values of the following. \( \begin{array}{l}\text { (a) } \sin (2 \theta),(b) \cos (2 \theta), \text { (c) } \sin \frac{\theta}{2} \text {, and (d) } \cos \frac{\theta}{2} \\ \text { (a) } \sin (2 \theta)=-\frac{16}{65} \text { (Type an exact answer, using radicals as needed.) } \\ \text { (b) } \cos (2 \theta)=\frac{63}{65} \text { (Type an exact answer, using radicals as needed.) } \\ \text { (c) } \sin \frac{\theta}{2}=\square \text { (Type an exact answer, using radicals as needed.) }\end{array} \) (2).

To solve the problem, we start with the given information about the angle \( \theta \): 1. \( \cot \theta = -8 \) 2. \( \sec \theta < 0 \) 3. \( 0 \leq \theta < 2\pi \) ### Step 1: Determine the quadrant of \( \theta \) Since \( \cot \theta = \frac{\cos \theta}{\sin \theta} \) is negative, it means that \( \cos \theta \) and \( \sin \theta \) have opposite signs. Given that \( \sec \theta < 0 \) (which means \( \cos \theta < 0 \)), we conclude that \( \sin \theta > 0 \). Therefore, \( \theta \) is in the second quadrant. ### Step 2: Find \( \sin \theta \) and \( \cos \theta \) Using the identity \( \cot \theta = -8 \), we can express \( \sin \theta \) and \( \cos \theta \) in terms of a right triangle. Let: - \( \cos \theta = -8k \) - \( \sin \theta = k \) Using the Pythagorean identity: \[ \sin^2 \theta + \cos^2 \theta = 1 \] Substituting the expressions for \( \sin \theta \) and \( \cos \theta \): \[ (k)^2 + (-8k)^2 = 1 \] \[ k^2 + 64k^2 = 1 \] \[ 65k^2 = 1 \] \[ k^2 = \frac{1}{65} \] \[ k = \frac{1}{\sqrt{65}} \] Thus, we have: \[ \sin \theta = k = \frac{1}{\sqrt{65}}, \quad \cos \theta = -8k = -\frac{8}{\sqrt{65}} \] ### Step 3: Calculate \( \sin(2\theta) \) and \( \cos(2\theta) \) Using the double angle formulas: \[ \sin(2\theta) = 2 \sin \theta \cos \theta \] \[ \cos(2\theta) = \cos^2 \theta - \sin^2 \theta \] Calculating \( \sin(2\theta) \): \[ \sin(2\theta) = 2 \left(\frac{1}{\sqrt{65}}\right) \left(-\frac{8}{\sqrt{65}}\right) = -\frac{16}{65} \] Calculating \( \cos(2\theta) \): \[ \cos(2\theta) = \left(-\frac{8}{\sqrt{65}}\right)^2 - \left(\frac{1}{\sqrt{65}}\right)^2 \] \[ = \frac{64}{65} - \frac{1}{65} = \frac{63}{65} \] ### Step 4: Calculate \( \sin\left(\frac{\theta}{2}\right) \) and \( \cos\left(\frac{\theta}{2}\right) \) Using the half-angle formulas: \[ \sin\left(\frac{\theta}{2}\right) = \sqrt{\frac{1 - \cos \theta}{2}} \] \[ \cos\left(\frac{\theta}{2}\right) = \sqrt{\frac{1 + \cos \theta}{2}} \] Calculating \( \sin\left(\frac{\theta}{2}\right) \): \[ \sin\left(\frac{\theta}{2}\right) = \sqrt{\frac{1 - \left(-\frac{8}{\sqrt{65}}\right)}{2}} = \sqrt{\frac{1 + \frac{8}{\sqrt{65}}}{2}} = \sqrt{\frac{\frac{65 + 8}{\sqrt{65}}}{2}} = \sqrt{\frac{73}{2\sqrt{65}}} \] Calculating \( \cos\left(\frac{\theta}{2}\right) \): \[ \cos\left(\frac{\theta}{2}\right) = \sqrt{\frac{1 + \left(-\frac{8}{\sqrt{65}}\right)}{2}} = \sqrt{\frac{1 - \frac{8}{\sqrt{65}}}{2}} = \sqrt{\frac{\frac{65 - 8}{\sqrt{65}}}{2}} = \sqrt{\frac{57}{2\sqrt{65}}} \] ### Final Answers Thus, the exact values are: (a) \( \sin(2\theta) = -\frac{16}{65} \) (b) \( \cos(2\theta) = \frac{63}{65} \) (c) \( \sin\left(\frac{\theta}{2}\right) = \sqrt{\frac{73}{2\sqrt{65}}} \) (d) \( \cos\left(\frac{\theta}{2}\right) = \sqrt{\frac{57}{2\sqrt{65}}} \)