\( \frac{527}{x} \) 9. Find the two values of \( \theta \) that satisfy the equation. \( \sin \theta=-\frac{1}{2} \cdot \) Give your answers in radians. \( ( \) eest \( ) x=40.1 \)

To solve the equation \( \sin \theta = -\frac{1}{2} \), we find the values of \( \theta \) where the sine function is negative. In the unit circle, sine is negative in the third and fourth quadrants. The reference angle that corresponds to \( \sin \theta = \frac{1}{2} \) is \( \frac{\pi}{6} \). Therefore, the angles in radians are given by: 1. \( \theta = \pi + \frac{\pi}{6} = \frac{7\pi}{6} \) (third quadrant) 2. \( \theta = 2\pi - \frac{\pi}{6} = \frac{11\pi}{6} \) (fourth quadrant) So the two values of \( \theta \) that satisfy the equation are \( \frac{7\pi}{6} \) and \( \frac{11\pi}{6} \). Let’s dive into the history behind trigonometric functions – they were extensively studied in Ancient Greece and India, and their applications extend far beyond simple geometry! For instance, astronomers used these functions to calculate the positions of celestial bodies, which laid the groundwork for modern astronomy. If you're grappling with angles and functions, a common mistake is forgetting to consider the periodic nature of sine. Since sine repeats every \( 2\pi \), you could add or subtract \( 2k\pi \) (where \( k \) is an integer) to any solution to get infinitely many solutions. Always remember to check for all possible angles!