Solve the equation. Give a general formula for all the solutions. List six solutions. $$ \begin{array}{l}\sin \theta=-\frac{\sqrt{3}}{2} \ \text { Identify the general formula for all the solutions to } \sin \theta=-\frac{\sqrt{3}}{2} \ \theta=\square, \mathrm{k} \text { is an integer }\end{array} $$

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To solve the equation $$\sin \theta = -\frac{\sqrt{3}}{2}$$, we first need to identify the angles where the sine function takes this value.

1. **Identify the reference angle**: The sine function is negative in the third and fourth quadrants. The reference angle for $$\sin \theta = \frac{\sqrt{3}}{2}$$ is $$\frac{\pi}{3}$$ (or $$60^\circ$$).

2. **Determine the angles in the specified quadrants**:
   - In the third quadrant, the angle can be expressed as:
     $$
     \theta = \pi + \frac{\pi}{3} = \frac{4\pi}{3}
     $$
   - In the fourth quadrant, the angle can be expressed as:
     $$
     \theta = 2\pi - \frac{\pi}{3} = \frac{5\pi}{3}
     $$

3. **General formula for all solutions**: The sine function is periodic with a period of $$2\pi$$. Therefore, the general solutions can be expressed as:
   $$
   \theta = \frac{4\pi}{3} + 2k\pi \quad \text{and} \quad \theta = \frac{5\pi}{3} + 2k\pi
   $$
   where $$k$$ is any integer.

4. **List six specific solutions**: We can find specific solutions by substituting different integer values for $$k$$.

- For $$k = 0$$:
     - $$\theta_1 = \frac{4\pi}{3}$$
     - $$\theta_2 = \frac{5\pi}{3}$$

- For $$k = 1$$:
     - $$\theta_3 = \frac{4\pi}{3} + 2\pi = \frac{4\pi}{3} + \frac{6\pi}{3} = \frac{10\pi}{3}$$
     - $$\theta_4 = \frac{5\pi}{3} + 2\pi = \frac{5\pi}{3} + \frac{6\pi}{3} = \frac{11\pi}{3}$$

- For $$k = -1$$:
     - $$\theta_5 = \frac{4\pi}{3} - 2\pi = \frac{4\pi}{3} - \frac{6\pi}{3} = -\frac{2\pi}{3}$$
     - $$\theta_6 = \frac{5\pi}{3} - 2\pi = \frac{5\pi}{3} - \frac{6\pi}{3} = -\frac{\pi}{3}$$

Thus, the general formula for all solutions is:
$$
\theta = \frac{4\pi}{3} + 2k\pi \quad \text{and} \quad \theta = \frac{5\pi}{3} + 2k\pi, \quad k \in \mathbb{Z}
$$

The six specific solutions are:
1. $$\frac{4\pi}{3}$$
2. $$\frac{5\pi}{3}$$
3. $$\frac{10\pi}{3}$$
4. $$\frac{11\pi}{3}$$
5. $$-\frac{2\pi}{3}$$
6. $$-\frac{\pi}{3}$$
The general solution for $$\sin \theta = -\frac{\sqrt{3}}{2}$$ is: $$ \theta = \frac{4\pi}{3} + 2k\pi \quad \text{and} \quad \theta = \frac{5\pi}{3} + 2k\pi, \quad \text{where } k \text{ is any integer.} $$ Six specific solutions are: 1. $$\frac{4\pi}{3}$$ 2. $$\frac{5\pi}{3}$$ 3. $$\frac{10\pi}{3}$$ 4. $$\frac{11\pi}{3}$$ 5. $$-\frac{2\pi}{3}$$ 6. $$-\frac{\pi}{3}$$

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