Question
Rose curves are characterized by equations of the form $r=a \cos (n \theta)$ or $r=a \sin (n \theta), a \neq 0 .$ If $n \neq 0$ is even, the rose has _____ petals. petals; if $n \neq \pm 1$ is odd, the rose has_____ petals.
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Step 1: The rose curves are given by the equations $r=a \cos (n \theta)$ or $r=a \sin (n \theta)$, where $a \neq 0$ and $n$ is an integer. Show more…
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Rose curves are characterized by equations of the form $r=a \cos (n \theta)$ or $r=a \sin (n \theta), a \neq 0 .$ If $n \neq 0$ is even, the rose has ___________ petals; if $n \neq \pm 1$ is odd, the rose has ___________ petals.
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THINK ABOUT IT How many petals do the rose curves given by $r=2\ \cos\ 4\theta$ and $r=2\ \sin\ 3\theta$ have? Determine the numbers of petals for the curves given by $r=2\ \cos\ n\theta$ and $r=2\ \sin\ n\theta$, where $n$ is a positive integer.
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Prove that a rose with an even number of petals is traced out exactly once as $\theta$ varies over the interval $0 \leq \theta<2 \pi$ and a rose with an odd number of petals is traced out exactly once as $\theta$ varies over the interval $0 \leq \theta<\pi$.
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