16. A portion of the graph of the polar function \( r=f(\theta) \), where \( f(\theta)=3 \sin (2 \theta) \), is shown in the polar coordinate system for \( a \leq \theta \leq b \). If \( 0 \leq a

Ask by Gordon Salinas. in the United States

Mar 12,2025

To determine the possible values for \( a \) and \( b \) in the polar function \( r = f(\theta) = 3 \sin(2\theta) \), we need to analyze the behavior of the function over the interval \( 0 \leq \theta < 2\pi \). ### Step 1: Understanding the Function The function \( f(\theta) = 3 \sin(2\theta) \) oscillates between \( -3 \) and \( 3 \) because the sine function oscillates between \( -1 \) and \( 1 \). The zeros of the function occur when \( \sin(2\theta) = 0 \), which happens at: \[ 2\theta = n\pi \quad \Rightarrow \quad \theta = \frac{n\pi}{2} \] for \( n = 0, 1, 2, 3, \ldots \) ### Step 2: Finding the Zeros The zeros in the interval \( [0, 2\pi) \) are: - \( \theta = 0 \) - \( \theta = \frac{\pi}{2} \) - \( \theta = \pi \) - \( \theta = \frac{3\pi}{2} \) ### Step 3: Analyzing the Intervals The function \( f(\theta) \) is positive when \( \sin(2\theta) > 0 \) and negative when \( \sin(2\theta) < 0 \). The sine function is positive in the intervals: - \( (0, \frac{\pi}{2}) \) - \( (\pi, \frac{3\pi}{2}) \) ### Step 4: Evaluating the Given Options Now, we will evaluate the provided options to see which intervals correspond to the positive values of \( f(\theta) \): 1. **Option (A)**: \( a = \frac{\pi}{4} \) and \( b = \frac{\pi}{2} \) - This interval \( \left[\frac{\pi}{4}, \frac{\pi}{2}\right] \) includes positive values since \( \frac{\pi}{4} \) is in the interval where \( f(\theta) > 0 \) and \( f(\frac{\pi}{2}) = 0 \). 2. **Option (B)**: \( a = \frac{\pi}{2} \) and \( b = \pi \) - This interval \( \left[\frac{\pi}{2}, \pi\right] \) includes \( f(\frac{\pi}{2}) = 0 \) and \( f(\theta) < 0 \) for \( \theta \in \left(\frac{\pi}{2}, \pi\right) \). 3. **Option (C)**: \( a = \frac{3\pi}{4} \) and \( b = 2\pi \) - This interval \( \left[\frac{3\pi}{4}, 2\pi\right] \) includes negative values since \( f(\frac{3\pi}{4}) < 0 \) and \( f(2\pi) = 0 \). 4. **Option (D)**: \( a = \frac{3\pi}{2} \) and \( b = 2\pi \) - This interval \( \left[\frac{3\pi}{2}, 2\pi\right] \) includes \( f(\frac{3\pi}{2}) = 0 \) and \( f(\theta) < 0 \) for \( \theta \in \left(\frac{3\pi}{2}, 2\pi\right) \). ### Conclusion The only option that corresponds to an interval where \( f(\theta) \) is positive is: **(A) \( a = \frac{\pi}{4} \) and \( b = \frac{\pi}{2} \)**.