QuestionSolve the equation. Give a general formula for all the solutions. List six solutions. $$\sin \theta=-\frac{1}{2}$$
Identify the general formula for all the solutions to $\sin \theta=-\frac{1}{2}$ based on the smaller angle. $\theta=$ $\square$ k is an integer (Simplify your answer. Use angle measureaz 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.)

Studdy Solution
List six specific solutions by substituting integer values for $k$ into the general formulas:
1. For $\theta = \frac{7\pi}{6} + 2k\pi$: - $k = 0$: $\theta = \frac{7\pi}{6}$ - $k = 1$: $\theta = \frac{7\pi}{6} + 2\pi = \frac{19\pi}{6}$ - $k = -1$: $\theta = \frac{7\pi}{6} - 2\pi = -\frac{5\pi}{6}$
2. For $\theta = \frac{11\pi}{6} + 2k\pi$: - $k = 0$: $\theta = \frac{11\pi}{6}$ - $k = 1$: $\theta = \frac{11\pi}{6} + 2\pi = \frac{23\pi}{6}$ - $k = -1$: $\theta = \frac{11\pi}{6} - 2\pi = -\frac{\pi}{6}$
The general formula for all solutions is: $\theta = \frac{7\pi}{6} + 2k\pi \quad \text{or} \quad \theta = \frac{11\pi}{6} + 2k\pi$
Six specific solutions are: $\frac{7\pi}{6}, \frac{19\pi}{6}, -\frac{5\pi}{6}, \frac{11\pi}{6}, \frac{23\pi}{6}, -\frac{\pi}{6}$