Use the following polar equation to answer the following questions. \( r=f(\theta)=3 \sin (2 \theta) \) name: \( R \) R -0\( ) \) 10. State the values of \( \theta \) where \( f(\theta)=0 \)

To find where \( f(\theta) = 0 \) in the polar equation \( r = 3 \sin(2\theta) \), we set \( 3 \sin(2\theta) = 0 \). This leads to solving \( \sin(2\theta) = 0 \). The sine function equals zero at multiples of \( \pi \), so we have \( 2\theta = n\pi \), where \( n \) is any integer. Therefore, the values of \( \theta \) can be expressed as \( \theta = \frac{n\pi}{2} \), resulting in angles like \( 0, \frac{\pi}{2}, \pi, \frac{3\pi}{2} \), and continuing infinitely in both directions! The beauty of this polar plot is that it creates a visually striking pattern. In this specific case, the function describes a rose curve with petals that appear at specific angles, and knowing where the sine function equals zero helps in graphically locating where these petals intersect the origin. So, grab a graphing tool and see those vibrant petals dance around the pole! 🌸📈