To find the polar form of the given parametric equations \( x = 4 \sin^2 \theta \) and \( y = 4 \sin \theta \cos \theta \), we can express them in terms of \( r \) and \( \theta \).
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Recall the relationships between Cartesian and polar coordinates: \[ x = r \cos \theta, \quad y = r \sin \theta \]
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Rewrite \( x \) and \( y \) in terms of polar coordinates: \[ x = 4 \sin^2 \theta \] and \[ y = 4 \sin \theta \cos \theta \]
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We can also use the double angle identity for sine: \[ y = 4 \sin \theta \cos \theta = 2 \sin(2\theta) \]
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Now, we can find \( r \) in terms of \( \theta \): From the equation \( y = 2 \sin(2\theta) \), we can express \( r \sin \theta \): \[ r \sin \theta = 2 \sin(2\theta) = 4 \sin \theta \cos \theta \] This gives us: \[ r = 4 \cos \theta \]
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Next, for \( x = 4 \sin^2 \theta \): We can express \( \sin^2 \theta \) using \( \sin \theta = \frac{y}{r} \): \[ x = 4 \left( \frac{y}{r} \right)^2 \]
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From these transformations, we derive \( r \) in terms of \( y \) or \( x \), but \( r = 4 \sin \theta \) simplifies to \( r = 4 \sin \theta \).
Thus, the polar form of the given equation is: \[ r = 4 \sin \theta \]
So the correct response is: \( r = 4 \sin \theta \).