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Can we use LaTeX to make the graph of $\rho = \sec(\theta)$?

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Have a look at this question: tex.stackexchange.com/a/65447/15925 . –  Andrew Swann Nov 24 '13 at 11:36
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4 Answers 4

You can use pgfplots to achieve this:

\documentclass[tikz]{standalone}

\usepackage{pgfplots}
    \pgfplotsset{compat=1.9}
    \usepgfplotslibrary{polar}

\begin{document}
    \begin{tikzpicture}
        \begin{polaraxis}
            \def\FREQUENCY{3}
            \addplot[red,domain=0:360,samples=360,smooth] (x,{30-8*sin(\FREQUENCY*x)});
            \addplot[dashed,domain=0:2*pi] (deg(x),30);
            \addplot[mark=*,only marks] coordinates {(0,0)};
        \end{polaraxis}
    \end{tikzpicture}
\end{document}

The source code is self-explanatory for anyone who have minimal experience in LaTeX.

Note: pgfplots by default uses degrees, for using radians you then need to convert it to degree via deg() function.

polar plot

And the \rho=\sec\theta (that's a pretty ugly function to plot it on the polar axis):

\documentclass[tikz]{standalone}

\usepackage{pgfplots}
    \pgfplotsset{compat=1.9}
    \usepgfplotslibrary{polar}

\begin{document}
    \begin{tikzpicture}
        \begin{polaraxis}
            \addplot[red,domain=-42:42,samples=360,smooth] (x,{sec(x)});
            \addplot[mark=*,only marks] coordinates {(0,0)};
        \end{polaraxis}
    \end{tikzpicture}
\end{document}

sec(theta)

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It is not \rho=\sec \theta. –  In PSTricks we trust Nov 24 '13 at 11:50
    
@DonutE.Knot: I've added the \rho=\sec\theta. It's a straight line indeed. –  m0nhawk Nov 24 '13 at 11:56

A recommended solution with PSTricks. \rho=\sec\theta represents the line x=1. So the correct curve should be like below.

\documentclass[border=12pt,pstricks]{standalone}

\usepackage{pst-plot}

\psset{runit=1.2cm,unit=\psrunit}
\pstVerb{/const 2 def}

\begin{document}
\begin{pspicture}(-3.5,-3.5)(3.5,3.5)
    \psaxes[axesstyle=polar,subticklinestyle=dashed,subticks=2,labelFontSize=\scriptstyle](3,3)
    \psplot[polarplot,algebraic=true,linecolor=red,linewidth=2pt,plotpoints=1000,yMaxValue=3.5,yMinValue=-3.5]{0}{TwoPi}{const/(cos(x))}
\end{pspicture}
\end{document}

enter image description here

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The polar with the corresponding cartesian plot:

\documentclass[border=12pt,pstricks]{standalone}
\usepackage{pst-plot}

\begin{document}
\begin{pspicture}(-3.5,-3.5)(3.5,3.5)
\psaxes[axesstyle=polar,subticklinestyle=dashed,subticks=2,labelFontSize=\scriptstyle](3,0)
 \psplot[polarplot,algebraic,linecolor=red,linewidth=2pt,plotpoints=500,
      yMaxValue=3.5]{Pi neg}{Pi}{1/(cos(x))}
 \psplot[algebraic,linecolor=blue,plotpoints=5000,yMaxValue=3.5]{Pi neg}{Pi}{1/(cos(x))}
\end{pspicture}

\end{document}

enter image description here

If you want to see the calculated points use showpoints:

\documentclass[border=12pt,pstricks]{standalone}
\usepackage{pst-plot}

\begin{document}
\begin{pspicture}(-3.5,-3.5)(3.5,3.5)
    \psaxes[axesstyle=polar,subticklinestyle=dashed,subticks=2,labelFontSize=\scriptstyle](3,0)
    \psplot[polarplot,algebraic,linecolor=red,linewidth=1.5pt,plotpoints=25,showpoints,
      yMaxValue=3.5]{Pi neg}{Pi}{1/(cos(x))}
\end{pspicture}

\end{document}

enter image description here

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My package xpicture plots polar curves (or more general, parametric curves).

For example

% Cardioide: r = 1+cos t
\SUMfunction{\ONEfunction}{\COSfunction}{\ffunction} % Define f(t)=1 + cos t
\POLARfunction{\ffunction}{\cardioide}  % Declare \cardioide as r=f(\phi)
% \degreespolarlabels              % Uncomment to label angles in degrees

\begin{center}
  \def\runitdivisions{2}              % 2 divisions of unity in the r-axis
  \setlength{\unitlength}{1.5cm}
  \begin{Picture}(-2.5,-2.5)(2.5,2.5)
    \polargrid{2}{16}  % Draw a polar grid for 0<r<2 and 16 divisions of circle
    \pictcolor{blue}\linethickness{1pt}
    \PlotParametricFunction[20]{\cardioide}{0}{\numberTWOPI} 
                                         % Draw \cardioide for 0<\phi<2\pi
  \end{Picture}

  $\rho=1+\cos\phi$
\end{center}

The cardioide curve

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