I'm wondering if anyone can give me tips or guidance on how to plot complicated polar curves without the use of a calculator. Most notably, I am trying to plot:
Based on a graphing calculator, I understand what this polar curve looks like, but I'm trying to learn and practice how to perform this procedure by hand. The odd angles $\dfrac{\pi}{4}$, $\dfrac{3\pi}{4}$ are difficult to me to evaluate in the cosine function. Please let me know if you have any tips to evaluating polar curves by hand.
Thanks in advanced,
Rusty
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$\begingroup$Plotting polar curves on Cartesian axes is just asking for a world of pain. Go to this site and download the graphic file. Print it out. Then you can easily plot curves by choosing various angles and calculating the corresponding radius. A Google image search for "polar coordinate paper" might also be helpful.
$\endgroup$ $\begingroup$Draw the graph of $y=\cos\tfrac12 x$ on a sheet of magic stretchy rubber. Then twist/stretch the rubber sheet to wrap it around the origin. The $x$ values become angles, and the $y$ values become distances from the origin.
If you don't have any magic rubber, you have to do this in your head. It's more difficult to visualize when some of the $y$ values are negative, so start by practicing with some examples where $y$ is always positive. Try $r = 3 + \sin100\theta$, for example. The function $y =3+\sin100x$ obviously just wiggles back and forth between $y=2$ and $y=4$. After the twisting/stretching/wrapping, you get something that wiggles back and forth between $r=2$ and $r=4$ as $\theta$ varies.
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