Solve the equation. Give a general formula for all the solutions. List six solutions. $$ \sin heta=-\frac{\sqrt{3}}{2} $$ Identify the general formula for all the solutions to $$ \sin heta=-\frac{\sqrt{3}}{2} $$ based on the smaller angle. $$ heta=\square, \mathrm{k} $$ 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.)

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Solve the equation. Give a general formula for all the solutions. List six solutions. $$ \sin 	heta=-\frac{\sqrt{3}}{2} $$ Identify the general formula for all the solutions to $$ \sin 	heta=-\frac{\sqrt{3}}{2} $$ based on the smaller angle. $$ 	heta=\square, \mathrm{k} $$ 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.)

UpStudy AI · Accepted Answer

To solve the equation $$ \sin \theta = -\frac{\sqrt{3}}{2} $$, we'll follow these steps:

### 1. **Determine the Reference Angle**
The reference angle $$ \alpha $$ where $$ \sin \alpha = \frac{\sqrt{3}}{2} $$ is:
$$
\alpha = \frac{\pi}{3}
$$

### 2. **Find the Quadrants Where Sine is Negative**
Sine is negative in the **third** and **fourth** quadrants. Therefore, the solutions within one full rotation ($$ 0 $$ to $$ 2\pi $$) are:
$$
\theta = \pi + \frac{\pi}{3} = \frac{4\pi}{3} \quad \text{(Third Quadrant)}
$$
$$
\theta = 2\pi - \frac{\pi}{3} = \frac{5\pi}{3} \quad \text{(Fourth Quadrant)}
$$

### 3. **Write the General Solution**
Since the sine function has a period of $$ 2\pi $$, the general solutions are:
$$
\theta = \frac{4\pi}{3} + 2\pi k \quad \text{or} \quad \theta = \frac{5\pi}{3} + 2\pi k \quad \text{where } k \text{ is any integer}
$$

### 4. **List Six Specific Solutions**
To list six solutions, we can choose different integer values for $$ k $$:

1. For $$ k = 0 $$:
   $$
   \theta = \frac{4\pi}{3}, \quad \frac{5\pi}{3}
   $$
   
2. For $$ k = 1 $$:
   $$
   \theta = \frac{4\pi}{3} + 2\pi = \frac{10\pi}{3}, \quad \frac{5\pi}{3} + 2\pi = \frac{11\pi}{3}
   $$
   
3. For $$ k = 2 $$:
   $$
   \theta = \frac{4\pi}{3} + 4\pi = \frac{16\pi}{3}, \quad \frac{5\pi}{3} + 4\pi = \frac{17\pi}{3}
   $$

### **Final Answer**

**General Formula:**
$$
\theta = \frac{4\pi}{3} + 2\pi k \quad \text{and} \quad \theta = \frac{5\pi}{3} + 2\pi k \quad \text{where } k \text{ is an integer}
$$

**Six Specific Solutions:**
$$
\frac{4\pi}{3}, \quad \frac{5\pi}{3}, \quad \frac{10\pi}{3}, \quad \frac{11\pi}{3}, \quad \frac{16\pi}{3}, \quad \frac{17\pi}{3}
$$
**General Solution:** $$ \theta = \frac{4\pi}{3} + 2\pi k \quad \text{and} \quad \theta = \frac{5\pi}{3} + 2\pi k \quad \text{where } k \text{ is an integer} $$ **Six Specific Solutions:** $$ \frac{4\pi}{3}, \quad \frac{5\pi}{3}, \quad \frac{10\pi}{3}, \quad \frac{11\pi}{3}, \quad \frac{16\pi}{3}, \quad \frac{17\pi}{3} $$

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