\begin{tabular}{|l|}\hline \\ 9. Find intervals of where \( r \) is positive and intervals \\ where \( r \) is negative. \\ \( r=4 \cos \theta \)\end{tabular}

To find the intervals where \( r \) is positive and where \( r \) is negative for the equation \( r = 4 \cos \theta \), we will follow these steps: 1. **Set the equation to find when \( r \) is positive:** \[ r > 0 \implies 4 \cos \theta > 0 \] This simplifies to: \[ \cos \theta > 0 \] 2. **Determine the intervals for \( \cos \theta > 0 \):** The cosine function is positive in the first and fourth quadrants. Therefore, the intervals for \( \theta \) where \( \cos \theta > 0 \) are: \[ 0 < \theta < \frac{\pi}{2} \quad \text{and} \quad \frac{3\pi}{2} < \theta < 2\pi \] 3. **Set the equation to find when \( r \) is negative:** \[ r < 0 \implies 4 \cos \theta < 0 \] This simplifies to: \[ \cos \theta < 0 \] 4. **Determine the intervals for \( \cos \theta < 0 \):** The cosine function is negative in the second and third quadrants. Therefore, the intervals for \( \theta \) where \( \cos \theta < 0 \) are: \[ \frac{\pi}{2} < \theta < \frac{3\pi}{2} \] ### Summary of Intervals: - **Positive intervals for \( r \):** \[ (0, \frac{\pi}{2}) \cup (\frac{3\pi}{2}, 2\pi) \] - **Negative intervals for \( r \):** \[ (\frac{\pi}{2}, \frac{3\pi}{2}) \] These intervals indicate where the value of \( r \) is positive and negative based on the angle \( \theta \).