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I'm trying to find patterns for polar curves. I just reviewed and feel comfortable with taking advantage of symmetry, but I still have trouble with some type of curves.

Limacons: Two types

1) inner loop

2) no inner loop

Is there a general formula that tells me whether there will be an inner loop? [tex]r=a\pm\cos\theta[/tex] and [tex]r=a\pm\sin\theta[/tex]

[tex]r=1+2\cos\theta[/tex] inner loop

[tex]r=1.5+\cos\theta[/tex] no inner loop

I tried to find a pattern myself, but I didn't find one.

[tex]r=1+2\cos\theta[/tex] inner loop, testing if b=even

[tex]r=a+3\cos\theta[/tex] inner loop, testing if a=k, b=odd

Now the reciprocal curves threw me off. I had forgotten about the range of cosecant, which is [tex][1,\infty)U(-\infty,-1][/tex], and I looped my curve back inwards, which is incorrect.

In general, when I'm graphing polar. Sine is symmetric with the y-axis, so the values of theta that I choose are from [tex]-\frac{\pi}{2}\leq\theta\leq\frac{\pi}{2}[/tex], and Cosine is symmetric with the x-axis, so I use [tex]0\leq\theta\leq\pi[/tex].

Now my main problem:

[tex]r=\csc\theta+2[/tex] (conchoid of Nicomedes)

Cosecant is also symmetric with the y-axis, so I choose my theta interval to be [tex]-\frac{\pi}{2}<\theta<\frac{\pi}{2}[/tex].

Limacons: Two types

1) inner loop

2) no inner loop

Is there a general formula that tells me whether there will be an inner loop? [tex]r=a\pm\cos\theta[/tex] and [tex]r=a\pm\sin\theta[/tex]

[tex]r=1+2\cos\theta[/tex] inner loop

[tex]r=1.5+\cos\theta[/tex] no inner loop

I tried to find a pattern myself, but I didn't find one.

[tex]r=1+2\cos\theta[/tex] inner loop, testing if b=even

[tex]r=a+3\cos\theta[/tex] inner loop, testing if a=k, b=odd

Now the reciprocal curves threw me off. I had forgotten about the range of cosecant, which is [tex][1,\infty)U(-\infty,-1][/tex], and I looped my curve back inwards, which is incorrect.

In general, when I'm graphing polar. Sine is symmetric with the y-axis, so the values of theta that I choose are from [tex]-\frac{\pi}{2}\leq\theta\leq\frac{\pi}{2}[/tex], and Cosine is symmetric with the x-axis, so I use [tex]0\leq\theta\leq\pi[/tex].

Now my main problem:

[tex]r=\csc\theta+2[/tex] (conchoid of Nicomedes)

Cosecant is also symmetric with the y-axis, so I choose my theta interval to be [tex]-\frac{\pi}{2}<\theta<\frac{\pi}{2}[/tex].

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