7. Find intervals of where \( f \) is increasing. \( r=4 \sin \theta \) inc:

1. Write the function as \[ f(\theta)=4\sin \theta. \] 2. Compute the derivative with respect to \(\theta\): \[ f'(\theta)=4\cos \theta. \] 3. The function is increasing when its derivative is positive: \[ 4\cos \theta>0 \quad \Longleftrightarrow \quad \cos \theta>0. \] 4. The cosine function is positive for angles in the intervals \[ \theta\in\left(-\frac{\pi}{2}+2\pi k,\, \frac{\pi}{2}+2\pi k\right), \quad k\in\mathbb{Z}. \] If we restrict \(\theta\) to the interval \([0,2\pi)\), these intervals become: \[ \theta\in\left(0,\, \frac{\pi}{2}\right) \quad \text{and} \quad \theta\in\left(\frac{3\pi}{2},\,2\pi\right). \] 5. Therefore, the function \( f(\theta)=4\sin \theta \) is increasing when \[ \theta\in\left\{ \theta: \cos \theta>0 \right\} \quad \text{or, equivalently,} \quad \theta\in\left(-\frac{\pi}{2}+2\pi k,\, \frac{\pi}{2}+2\pi k\right) \text{ for any integer } k. \]