Solve the equation. Give a general formula for all the solutions. List six solutions. \[ \sin \theta=\frac{\sqrt{3}}{2} \] Identify the general formula for all the solutions to \( \sin \theta=\frac{\sqrt{3}}{2} \) based on the smaller angle, \( \theta=\square, k \) is an integer (Simplify your answer. Use angle measures greater than or equal to 0 and less than \( 2 \pi \). Type an exact answer, using \( \pi \) as needed. Use integers or fractions for any numbers in the expression. Type an expression using k as the variable.)

To solve the equation \( \sin \theta = \frac{\sqrt{3}}{2} \), we first identify the angles where the sine function takes this value. The two primary angles in the range \( [0, 2\pi) \) are \( \theta = \frac{\pi}{3} \) and \( \theta = \frac{2\pi}{3} \). The general formula for all solutions can be expressed as: \[ \theta = \frac{\pi}{3} + 2k\pi \quad \text{or} \quad \theta = \frac{2\pi}{3} + 2k\pi, \quad k \in \mathbb{Z} \] Now, let's list six specific solutions by choosing integer values for \( k \): 1. For \( k = 0 \): \( \frac{\pi}{3} \) and \( \frac{2\pi}{3} \) 2. For \( k = 1 \): \( \frac{\pi}{3} + 2\pi = \frac{7\pi}{3} \) and \( \frac{2\pi}{3} + 2\pi = \frac{8\pi}{3} \) 3. For \( k = -1 \): \( \frac{\pi}{3} - 2\pi = -\frac{5\pi}{3} \) (but we can keep it positive, so we can use equivalent positive angle: \( \frac{4\pi}{3} \)) and \( \frac{2\pi}{3} - 2\pi = -\frac{4\pi}{3} \) (equivalent angle: \( \frac{2\pi}{3} \)) The six specific solutions are: 1. \( \frac{\pi}{3} \) 2. \( \frac{2\pi}{3} \) 3. \( \frac{7\pi}{3} \) 4. \( \frac{8\pi}{3} \) 5. \( \frac{4\pi}{3} \) 6. \( \frac{2\pi}{3} \) (this one repeats) So, adjusting for distinct values we get unique solutions of: 1. \( \frac{\pi}{3} \) 2. \( \frac{2\pi}{3} \) 3. \( \frac{7\pi}{3} \) 4. \( \frac{8\pi}{3} \) 5. \( \frac{4\pi}{3} \) 6. \( \frac{5\pi}{3} \) (as the next valid solution) In summary, the unique solutions include angles that showcase the periodic nature of sine!