Solve the equation. Give a general formula for all the solutions. List six solutions. \( \sin \theta=\frac{1}{2} \) \( \theta=\square, k \) is an integer Identify the general formula for all the solutions to \( \sin \theta=\frac{1}{2} \) based on the smaller angle. (Simplify your answer. Use angle measures greater than or equal to 0 and less than \( 2 \pi \). Type an exact answer, using 1 . \( \pi \) as nes needed. Use integers or fractions for any numbers in the expression. Type an expression using k as the variable.)

To solve the equation \( \sin \theta = \frac{1}{2} \), we'll find all angles \( \theta \) that satisfy this condition. ### Step 1: Identify the Reference Angle The reference angle where \( \sin \theta = \frac{1}{2} \) is: \[ \theta = \frac{\pi}{6} \] ### Step 2: Determine All Solutions in the Interval \([0, 2\pi)\) The sine function is positive in the first and second quadrants. Therefore, the solutions within one full rotation (\(0 \leq \theta < 2\pi\)) are: \[ \theta = \frac{\pi}{6} \quad \text{and} \quad \theta = \frac{5\pi}{6} \] ### Step 3: Generalize for All Solutions The sine function has a period of \(2\pi\), meaning the solutions repeat every \(2\pi\) radians. Therefore, the general formulas for all solutions are: \[ \theta = \frac{\pi}{6} + 2k\pi \] \[ \theta = \frac{5\pi}{6} + 2k\pi \] where \( k \) is any integer. ### Step 4: List Six Specific Solutions By substituting \( k = 0, 1, 2 \) into the general formulas, we obtain six specific solutions: 1. For \( k = 0 \): \[ \theta = \frac{\pi}{6} \] \[ \theta = \frac{5\pi}{6} \] 2. For \( k = 1 \): \[ \theta = \frac{\pi}{6} + 2\pi = \frac{13\pi}{6} \] \[ \theta = \frac{5\pi}{6} + 2\pi = \frac{17\pi}{6} \] 3. For \( k = 2 \): \[ \theta = \frac{\pi}{6} + 4\pi = \frac{25\pi}{6} \] \[ \theta = \frac{5\pi}{6} + 4\pi = \frac{29\pi}{6} \] ### Summary - **General Formulas:** \[ \theta = \frac{\pi}{6} + 2k\pi \quad \text{and} \quad \theta = \frac{5\pi}{6} + 2k\pi \quad \text{where } k \text{ is an integer} \] - **Six Specific Solutions:** \[ \theta = \frac{\pi}{6}, \quad \frac{5\pi}{6}, \quad \frac{13\pi}{6}, \quad \frac{17\pi}{6}, \quad \frac{25\pi}{6}, \quad \frac{29\pi}{6} \]