[Tex/LaTex] Drawing and Labeling a Triangle for a Demonstration of the Double-Angle Formula of Cosines

pgfplotstikz-pgf

I have a display of two triangles, but the triangle drawn in red is unintentional. TikZ is misinterpreting the coordinates for point C. The polar coordinates for C is given as \coordinate (C) at (120:5);. TikZ is plotting the point in Quadrant III.

Also, TikZ is misinterpreting the positions at which I want the label for the vertex A and the labels for the angles to be typeset. The command to place A is

\node at ({2.5mm*sqrt(2)/2},{-2.5mm*sqrt(2)/2}){$A$};

and the commands to place $\theta$ are

\coordinate (label_for_theta) at (60:4mm);

\node[font=\footnotesize] at (label_for_theta){$\theta$};.

It seems to me that TikZ is placing these labels using the canvas coordinate system.

\documentclass{amsart}
\usepackage{amsmath}
\usepackage{amsfonts}

\usepackage{tikz}
\usetikzlibrary{calc,angles,positioning,intersections,quotes,decorations.markings,backgrounds,patterns}

\usepackage{pgfplots}
\pgfplotsset{compat=1.11}


\begin{document}


\begin{tikzpicture}

\begin{axis}[width=5in,axis equal image,
    axis lines=middle,
    xmin=-5,xmax=5,
    ymin=-1.5,ymax=5,
    restrict y to domain=-1.5:5,
    enlargelimits={abs=0.5cm},
    xtick={\empty},ytick={\empty},
    axis line style={latex-latex},
    xlabel=$x$,ylabel=$y$,
    xlabel style={at={(ticklabel* cs:1)},anchor=north west},
    ylabel style={at={(ticklabel* cs:1)},anchor=south west}
]

%A triangle is drawn on the Cartesian plane. One side of the triangle is along
%the positive x-axis, and another side of the triangle is drawn in Quadrant II.
\coordinate (A) at (0,0);
\coordinate (B) at (3.5,0);
\coordinate (C) at (120:5);
\draw[red] (A) -- (B) -- (C) -- cycle;

%The labels for A and B are typeset.
\node at ({2.5mm*sqrt(2)/2}:{-2.5mm*sqrt(2)/2}){$A$};
\node at (3.5,-2.5mm){$B$};

%The point C' = C is placed using Cartesian coordinates.
\coordinate (C') at (-2.5,{5*sin(120)});
\draw[fill] (C') circle (1.5pt);
\draw (A) -- (C');
\draw (B) -- (C');
%The label for C' = C is typeset.
\coordinate (label_C_left) at ($(C')!-4mm!(B)$);
\coordinate (label_C_right) at ($(C')!-4mm!(A)$);
\coordinate (label_C) at ($(label_C_left)!0.5!(label_C_right)$);
\node[blue] at ($(C')!2.5mm!(label_C)$){$C$};

%Angles are drawn for $\theta$ and its supplement.
\draw[draw=blue] (A) ++(120:4mm) arc (120:0:4mm);
\coordinate (label_for_theta) at (60:6.5mm);
\node[font=\footnotesize] at (label_for_theta){$\theta$};
\draw[draw=blue,dash dot] (A) ++(180:6mm) arc (180:120:6mm);
\coordinate (label_for_supplement_to_theta) at (150:8.5mm);
\node[font=\footnotesize] at (label_for_supplement_to_theta){$\pi - \theta$};


%A right-angle mark is drawn.
\coordinate (U) at ($(-2.5,0)!3mm!45:(A)$);
\draw[dash dot] (U) -- ($(-2.5,0)!(U)!(A)$);
\draw[dash dot] (U) -- ($(-2.5,0)!(U)!(C')$);


\draw[dashed] (C') -- (-2.5,0);

%Braces indicating the distances that C' is from the axes are typeset. To give
%them the appearance of being typeset over the axes, they are first typeset
%in white with a line width of 2pt, which is 10 times the thickness of the
%brace that is actually typeset.
\draw[draw=white,line width=4pt,decorate,decoration={brace,raise=5pt,amplitude=5pt}] (-2.5,0) -- (C');
\draw[decorate,decoration={brace,raise=5pt,amplitude=5pt}] (-2.5,0) -- (C');
\draw[draw=white,line width=4pt,decorate,decoration={brace,raise=5pt,amplitude=5pt,mirror}] (-2.5,0) -- (A);
\draw[decorate,decoration={brace,raise=5pt,amplitude=5pt,mirror}] (-2.5,0) -- (A);


\coordinate (label_for_5_sin_theta) at ($({5*cos(120)},{2.5*sin(120)}) + (-2.5mm-10pt,0pt)$);
\node[anchor=east] at (label_for_5_sin_theta){$r\sin\theta$};
\coordinate (label_for_5_cos_theta) at (-1.25,-2.5mm-10pt);
\node at (label_for_5_cos_theta){$r\cos\theta$};

\end{axis}
\end{tikzpicture}


\end{document}

Best Answer

You cannot straightforwardly use dimensions when specifying coordinates within an axis environment. Since the specification for C doesn't use a dimension, it is correctly placed, but other coordinates are misplaced relative to the intended origin.

To see that this is not an effect of TikZ itself, simply remove the axis environment and typeset the remaining code within the raw tikzpicture environment.

The reason this does not work in the axis environment is because there is, in general, no one-one correspondence between the coordinate systems used by TikZ and those configured by pgfplots. See page 350, section 4.27 TikZ Interoperability for details.

One option is to simply dispense with pgfplots as you are not really using it except to draw the axes:

\documentclass[tikz,border=10pt,multi]{standalone}
\usepackage{amsmath}
\usepackage{amsfonts}
\usetikzlibrary{calc,arrows.meta,positioning}
\usepackage{pgfplots}
\pgfplotsset{compat=1.11}
\begin{document}

\begin{tikzpicture}
  \draw [{Latex[]}-{Latex[]}] (-5.5,0) -- (5.5,0) node [below right] {$x$};
  \draw [{Latex[]}-{Latex[]}] (0,-1.5) -- (0,5.5) node [above right] {$y$};

  %A triangle is drawn on the Cartesian plane. One side of the triangle is along
  %the positive x-axis, and another side of the triangle is drawn in Quadrant II.
  \coordinate (A) at (0,0);
  \coordinate (B) at (3.5,0);
  \coordinate (C) at (120:5);
  \draw[draw=red] (A) node [below right] {$A$} -- (B) -- (C) -- cycle;

  %The label for B is typeset.
  \node at (3.5,-2.5mm){$B$};

  %The point C' = C is placed using Cartesian coordinates.
  \coordinate (C') at (-2.5,{5*sin(120)});
  \draw[fill] (C') circle (1.5pt);
  \draw (A) -- (C');
  \draw (B) -- (C');
  %The label for C' = C is typeset.
  \coordinate (label_C_left) at ($(C')!-4mm!(B)$);
  \coordinate (label_C_right) at ($(C')!-4mm!(A)$);
  \coordinate (label_C) at ($(label_C_left)!0.5!(label_C_right)$);
  \node[blue] at ($(C')!2.5mm!(label_C)$){$C$};

  %Angles are drawn for $\theta$ and its supplement.
  \draw[draw=blue] (A) ++(120:4mm) arc (120:0:4mm);
  \coordinate (label_for_theta) at (60:6mm);
  \node[font=\footnotesize] at (label_for_theta){$\theta$};
  \draw[draw=blue,dash dot] (A) ++(180:6mm) arc (180:120:6mm);
  \coordinate (label_for_supplement_to_theta) at (150:9.5mm);
  \node[font=\footnotesize] at (label_for_supplement_to_theta){$\pi - \theta$};

  %A right-angle mark is drawn.
  \coordinate (U) at ($(-2.5,0)!3mm!45:(A)$);
  \draw[dash dot] (U) -- ($(-2.5,0)!(U)!(A)$);
  \draw[dash dot] (U) -- ($(-2.5,0)!(U)!(C')$);

  \draw[dashed] (C') -- (-2.5,0);

  %Braces indicating the distances that C' is from the axes are typeset. To give
  %them the appearance of being typeset over the axes, they are first typeset
  %in white with a line width of 2pt, which is 10 times the thickness of the
  %brace that is actually typeset.
  \draw[draw=white,line width=4pt,decorate,decoration={brace,raise=5pt,amplitude=5pt}] (-2.5,0) -- (C');
  \draw[decorate,decoration={brace,raise=5pt,amplitude=5pt}] (-2.5,0) -- (C');
  \draw[draw=white,line width=4pt,decorate,decoration={brace,raise=5pt,amplitude=5pt,mirror}] (-2.5,0) -- (A);
  \draw[decorate,decoration={brace,raise=5pt,amplitude=5pt,mirror}] (-2.5,0) -- (A);

  \coordinate (label_for_5_sin_theta) at ($({5*cos(120)},{2.5*sin(120)}) + (-2.5mm-10pt,0pt)$);
  \node[anchor=east] at (label_for_5_sin_theta){$r\sin\theta$};
  \coordinate (label_for_5_cos_theta) at (-1.25,-2.5mm-10pt);
  \node at (label_for_5_cos_theta){$r\cos\theta$};
\end{tikzpicture}
\end{document}

Another option is to specify the coordinates differently. Although polar coordinates using absolute dimensions do not work within the axis environment you've defined, specifying them without using absolute dimensions does:

\documentclass[tikz,border=10pt]{standalone}
\usepackage{amsmath}
\usepackage{amsfonts}
\usetikzlibrary{calc,angles,positioning,intersections,quotes,decorations.markings,backgrounds,patterns}
\usepackage{pgfplots}
\pgfplotsset{compat=1.11}
\begin{document}
\begin{tikzpicture}
  \begin{axis}[width=5in,axis equal image,
    axis lines=middle,
    xmin=-5,xmax=5,
    ymin=-1.5,ymax=5,
    restrict y to domain=-1.5:5,
    enlargelimits={abs=0.5cm},
    xtick={\empty},ytick={\empty},
    axis line style={latex-latex},
    xlabel=$x$,ylabel=$y$,
    xlabel style={at={(ticklabel* cs:1)},anchor=north west},
    ylabel style={at={(ticklabel* cs:1)},anchor=south west}
    ]

    %A triangle is drawn on the Cartesian plane. One side of the triangle is along
    %the positive x-axis, and another side of the triangle is drawn in Quadrant II.
    \draw [draw= red] (0,0) coordinate (A) node [below right] {$A$} -- (3.5,0) coordinate (B) node [below] {$B$} -- (120:5) coordinate (C) -- cycle;

    %The point C' = C is placed using Cartesian coordinates.
    \draw (A) --  (-2.5,{5*sin(120)}) coordinate (C') -- (B);
    \draw[fill] (C') circle (1.5pt);
    %The label for C' = C is typeset.
    \coordinate (label_C_left) at ($(C')!-4mm!(B)$);
    \coordinate (label_C_right) at ($(C')!-4mm!(A)$);
    \coordinate (label_C) at ($(label_C_left)!0.5!(label_C_right)$);
    \node[blue] at ($(C')!2.5mm!(label_C)$){$C$};

    %Angles are drawn for $\theta$ and its supplement.
    \draw[draw=blue] (120:0.4) arc (120:0:0.4);
    \coordinate (label_for_theta) at (60:0.4);
    \node[font=\footnotesize, anchor=south west] at (label_for_theta){$\theta$};
    \draw[draw=blue,dash dot] (180:.6) arc (180:120:.6);
    \coordinate (label_for_supplement_to_theta) at (150:.6);
    \node[font=\footnotesize, anchor=south east] at (label_for_supplement_to_theta){$\pi - \theta$};

    %A right-angle mark is drawn.
    \draw[dash dot] ($(-2.5,0)!3mm!45:(A)$) coordinate (U) -- ($(-2.5,0)!(U)!(A)$);
    \draw[dash dot] (U) -- ($(-2.5,0)!(U)!(C')$);

    \draw[dashed] (C') -- (-2.5,0);

    %Braces indicating the distances that C' is from the axes are typeset. To give
    %them the appearance of being typeset over the axes, they are first typeset
    %in white with a line width of 2pt, which is 10 times the thickness of the
    %brace that is actually typeset.
    \draw[draw=white,line width=4pt,decorate,decoration={brace,raise=5pt,amplitude=5pt}] (-2.5,0) -- (C');
    \draw[decorate,decoration={brace,raise=5pt,amplitude=5pt}] (-2.5,0) -- (C');
    \draw[draw=white,line width=4pt,decorate,decoration={brace,raise=5pt,amplitude=5pt,mirror}] (-2.5,0) -- (A);
    \draw[decorate,decoration={brace,raise=5pt,amplitude=5pt,mirror}] (-2.5,0) -- (A);

    \coordinate (label_for_5_sin_theta) at ($({5*cos(120)},{2.5*sin(120)}) + (-2.5mm-10pt,0pt)$);
    \node[anchor=east] at (label_for_5_sin_theta){$r\sin\theta$};
    \coordinate (label_for_5_cos_theta) at (-1.25,-2.5mm-10pt);
    \node at (label_for_5_cos_theta){$r\cos\theta$};

  \end{axis}
\end{tikzpicture}
\end{document}

It is true that you can usually use dimensions to say e.g. (60:6mm) in an ordinary tikzpicture environment.

Related Solutions

[Tex/LaTex] Rotate a node but not its content: the case of the ellipse decoration

The source of the difficulty is that ellipses are constructed in a particular way in TikZ. They are paths that start from the x-axis and proceed counter-clockwise around their centre. The vast majority of the time, the exact parametrisation doesn't matter. You appear to have found the one situation where it does!

In the actual question, you only want to be able to mirror the ellipse, and so draw it starting from the negative x-axis (the title of the question suggests a more flexible approach). That's actually not too hard since we can exploit the symmetry of the ellipse. The key is to provide it with a negative x-radius, since then it will start from the negative x-axis (and proceed clockwise, but we could correct for that by negating the y-radius as well). To do this, we interrupt the call from the node shape to the drawing command and flip the sign of the x-radius. The simplest way to do this is to redefine the \pgfpathellipse macro to do the negation and then call the original macro. The following code does this.

\documentclass{article}
\usepackage{tikz}
\usetikzlibrary{decorations,shapes,decorations.markings}
\makeatletter
\let\origpgfpathellipse=\pgfpathellipse
\def\revpgfpathellipse#1#2#3{%
  #2%
  \pgf@xa=-\pgf@x
  \origpgfpathellipse{#1}{\pgfqpoint{\pgf@xa}{0pt}}{#3}}
\makeatother

\tikzset{
  reversed ellipse/.style={
    ellipse,
    reverse the ellipse%
  },
  reverse the ellipse/.code={
    \let\pgfpathellipse=\revpgfpathellipse
  }
}
\begin{document}
\begin{tikzpicture}
\node[ellipse,
  draw,
  postaction={
    decorate,
    decoration={
      markings,
      mark=at position 1 with {
        \arrow[line width=5pt,blue]{>}
      }
    }
  }
] at (0,0) {hello world};

\node[reversed ellipse,
  draw,
  postaction={
    decorate,
    decoration={
      markings,
      mark=at position 1 with {
        \arrow[line width=5pt,blue]{>}
      }
    }
  }
] at (0,-2) {hello world};
\end{tikzpicture}

\end{document}

Here's the result:

rotated ellipse

(the arrow got clipped, but you can see where it lies)

[Tex/LaTex] Drawing and labeling asymptotes

You can use a node to put x=1 and y=1. Further, to draw a horizontal line at y=1, you can use \addplot: like

\addplot[samples=200,dashed,latex-latex,domain=-17:17] {1}node[above,pos=0]{$y=1$};

Full code:

\documentclass[10pt]{amsart}
%\usepackage{tikz}      %% These are all not needed
%\usetikzlibrary{calc,angles,positioning,intersections,quotes,decorations.markings}
%\usepackage{tkz-euclide}
%\usetkzobj{all}
\usepackage{pgfplots}
\pgfplotsset{compat=1.11}

%\usepackage{amsmath}
%\usepackage{amsfonts}
%\usepackage{amssymb}
%\usepackage{amsthm}

\begin{document}

\begin{tikzpicture}
\begin{axis}[width=4in,axis equal image,
    axis lines=middle,
    xmin=-15,xmax=15,
    xlabel=$x$,ylabel=$y$,
    ymin=-10,ymax=10,
    restrict y to domain=-12:12,
    enlargelimits={abs=1cm},
    axis line style={latex-latex},
    ticklabel style={font=\tiny,fill=white},
    xtick={-3},
    every axis y label/.style=
         {at={(ticklabel cs:1.02,-12pt)},rotate=0,anchor=west},
    every axis x label/.style={
         at={(xticklabel cs:1.02,-8pt)},anchor=south},
]
/pgfplots/xlabel shift={10pt};
\addplot[samples=250,domain=-15:15] {(x+2)/(x-1)};
\draw[dashed,shorten <=2ex, shorten >=2ex,latex-latex] ({{1,0}}|-{{axis description cs:1,1}}) -- ({{1,0}}|-{{axis description cs:1,0}})node[right,pos=0.95]{$x=1$};
\draw [fill=white] (-3,0.25) circle [radius=1.5pt] node[left]{};
\addplot[samples=200,dashed,latex-latex,domain=-17:17] {1}node[above,pos=0]{$y=1$};
\end{axis}
\end{tikzpicture}
\vskip0.25in


\begin{tikzpicture}
\begin{axis}[width=4in,axis equal image,
    axis lines=middle,
    xmin=-15,xmax=15,
    xlabel=$x$,ylabel=$y$,
    ymin=-10,ymax=10,
    restrict y to domain=-12:12,
    enlargelimits={abs=1cm},
    axis line style={latex-latex},
    ticklabel style={font=\tiny,fill=white},
    xtick={-1},ytick=\empty,
    every axis y label/.style=
         {at={(ticklabel cs:1.02,0pt)},rotate=0,},
    every axis x label/.style={
         at={(xticklabel cs:1.02,-8pt)},anchor=south},
]
/pgfplots/xlabel shift={10pt};
\addplot[samples=251,domain=-14:16] {(x+1)/(x-1)};
\draw[dashed,shorten <=2ex, shorten >=2ex,latex-latex] ({{1,0}}|-{{axis description cs:1,1}}) -- ({{1,0}}|-{{axis description cs:1,0}})node[right,pos=0.95]{$x=1$};
\addplot[samples=200,dashed,latex-latex,domain=-17:17] {1}node[above,pos=0]{$y=1$};

\end{axis}
\end{tikzpicture}

\end{document}

enter image description here

Best Answer

Related Solutions

[Tex/LaTex] Rotate a node but not its content: the case of the ellipse decoration

[Tex/LaTex] Drawing and labeling asymptotes

Related Question