Tikz package. Trefoil.

Question

How to plot a trefoil like this in Tikzpicture?

Answer 1

As I alluded to, you basically only have to copy an example from section 22.5 in the manual for TikZ and replace the equations with the ones you want. In the first example note that the r indicates that radians should be used, by default the trigonometric functions in pgf uses degrees.

I also added an example with pgfplots, which is convenient if you need axes.

\documentclass[border=4mm]{standalone} 
\usepackage{pgfplots} % also loads tikz
\begin{document}              
\begin{tikzpicture}
\draw[domain=0:pi,variable=\t,samples=200]
plot ({cos(\t r)^2*(-1 + 4*sin(\t r)^2)},{cos(\t r)*sin(\t r)*(-1 + 4*sin(\t r)^2)});
\end{tikzpicture}

\begin{tikzpicture}
\begin{axis}[
  axis lines=middle,
  trig format=rad,
  domain=0:pi,
  samples=200,
  width=4cm,
  axis equal]
\addplot [black] ({cos(x)^2*(-1 + 4*sin(x)^2)},{cos(x)*sin(x)*(-1 + 4*sin(x)^2)});
\end{axis}
\end{tikzpicture}
\end{document}