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
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.