\documentclass[border=10pt]{standalone}
%%%<
\usepackage{verbatim}
%%%>
\begin{comment}
:Title: Polar plot of a sine function with factor in argument
:Tags: 2D;Functions;Polar Plots
:Author: Stefan Kottwitz
:Slug: sine-plot
The sine function is a well known periodic function. Choose an argument
factor, which doesn't divide the period well, and you can see a complicated
path around the origin.
This plot was originally made for my blog post on TikZ.de:
http://tikz.de/periodisch/
See also the other polar sine plot examples there.
\end{comment}
\usepackage{pgfplots}
\usepgfplotslibrary{polar,colormaps}
\begin{document}
\begin{tikzpicture}
\begin{polaraxis}[
domain = -14400:14400,
samples = 3000,
colormap/cool,
hide axis
]
\addplot[no markers, mesh, opacity=0.5] {1-sin(40*x/39};
\end{polaraxis}
\end{tikzpicture}
\end{document}