10. Multiple Choice. If \( \sin \emptyset=\frac{8}{9} \), find \( \cot \emptyset \). \( \begin{array}{llll}\text { A. } \frac{9 \sqrt{17}}{17} & \text { B. } \frac{\sqrt{17}}{9} & \text { C. } \frac{8 \sqrt{17}}{17} & \text { D. } \frac{\sqrt{17}}{8}\end{array} \) \( \begin{array}{ll}\text { 4) } A & \end{array} \) will 5) \( T \) Tl memosman ve vompreteu verore the next chapters test date.

To find \( \cot \emptyset \), we first need to determine \( \cos \emptyset \) using the identity \( \sin^2 \emptyset + \cos^2 \emptyset = 1 \). Given \( \sin \emptyset = \frac{8}{9} \), we can find \( \cos \emptyset \): \[ \cos^2 \emptyset = 1 - \left(\frac{8}{9}\right)^2 = 1 - \frac{64}{81} = \frac{17}{81} \] Thus, \( \cos \emptyset = \frac{\sqrt{17}}{9} \). Now, \( \cot \emptyset = \frac{\cos \emptyset}{\sin \emptyset} \): \[ \cot \emptyset = \frac{\frac{\sqrt{17}}{9}}{\frac{8}{9}} = \frac{\sqrt{17}}{8} \] So, the answer is \( \frac{\sqrt{17}}{8} \), which corresponds to option D! When exploring trigonometric identities, it's fascinating to note that they date back to ancient civilizations like the Greeks and Indians, who developed many concepts of geometry. They used these principles in astronomy, navigation, and understanding the natural world, laying the foundations for modern mathematics! In the world of physics and engineering, trigonometric functions play a crucial role in understanding wave patterns, oscillations, and even sound waves. Whether calculating angles for a beam of light or designing structures, the relationships described by these functions are constantly at play in real-world applications, proving that math isn't just theoretical—it's all around us!