MHB What are the key characteristics of polar curves like petals and limacons?

[Eissa12]

View attachment 8979A) Find all values on [0,2pie) such that (thita0) produces the tip of a petal (maximum magnitude of r) all values for which r=0, and sketch a graph?

a) r = 5 sin 2 (thita0)

a) r = 5 sin 3 (thita0)

a) r = 5 sin 4 (thita0)

B) considering what you can observe in the previous graphs, what are general guidelines for number, length and position of petals for a general rose curve: r = a sin b (thita0).

c) A polar curve in the frome r = d + a sin (thita0) is called a limacon and has several distinct variations, including a cardioid and a limacon with an inner loop. Create general guidelines for when these variations occur and explain what causes them to occur?

 

[skeeter]

tip of the petal $\implies$ $\cos(2\theta) = \pm 1$ ...

$0 \le \theta < 2\pi \implies 0 \le 2\pi < 4\pi$$r = 5\cos(2\theta) \implies \cos(2\theta) = \pm 1 \implies 2\theta \in \left\{ 0, \pi, 2\pi, 3\pi \right\} \implies \theta \in \left\{0, \dfrac{\pi}{2}, \pi, \dfrac{3\pi}{2} \right\}$

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