Graphing trigonometric functions, axes labeled with tick marks

Question

I am providing the code for graphing three trigonometric functions. (In the preamble, I had to put a % before the command \usetikzlibrary{calc,angles,quotes} because LaTeX could not compile it with the command. It gave me the error can't find file 'tikzlibraryangles.code.tex'. MikTeX2.9 is compiling the code.) How do I add the labels "$\scriptstyle{y = \sin(x)}," "$\scriptstyle{y = \cos(x)}," and "$\scriptstyle{y = \tan(x)}" directly under the y-axis to the corresponding graphs? Arrowheads should be at both ends of the axes. They are currently at one end. Is there a code for putting labels (in nodes) for the tick marks over other features of the graphs? How does the code instruct TikZ to marks the tick marks in half-unit increments? How do I get the code to instruct TikZ to mark the tick marks in increments of integral multiples of \pi/2? In the graphs of the tangent function, the asymptotes are solid lines. How do I get dotted lines? In the code for graphing the tangent function, there is "scaled x ticks={real:3.1415}, xtick scale label code/.code={}". What does "{real:3.1415}" and "code/.code={}" instruct TikZ to draw on the x-axis?

\documentclass{article}
\usepackage{amsmath}
\usepackage{tikz}
%\usetikzlibrary{calc,angles,quotes}
\usepackage{pgfplots}


\pgfkeys{/pgfplots/Axis Style/.style={
    width=13.5cm, height=5cm,
    axis x line=center, 
    axis y line=middle, 
    samples=100,
    ymin=-1.5, ymax=1.5,
    xmin=-7.0, xmax=7.0,
    domain=-2*pi:2*pi
}}


\begin{document}

\begin{tikzpicture}
\begin{axis}[
    Axis Style,
    xtick={
        -6.28318, -4.7123889, -3.14159, -1.5708,
        1.5708, 3.14159, 4.7123889, 6.28318
    },
    xticklabels={
        $-2\pi$, $-3\pi/2$, $-\pi$, $\pi/2$,
        $\pi/2$, $\pi$, $3\pi/2$, $2\pi$
    }
]
\addplot [mark=none, ultra thick, blue] {sin(deg(x))};
\end{axis}
\end{tikzpicture}

\begin{tikzpicture}
\begin{axis}[
    Axis Style,
    xtick={
        -6.28318, -4.7123889, -3.14159, -1.5708,
        1.5708, 3.14159, 4.7123889, 6.28318
    },
    xticklabels={
        $-2\pi$, $-3\pi/2$, $-\pi\hspace{0.30cm}$, $\pi/2$,
        $\pi/2$, $\pi\hspace{0.10cm}$, $3\pi/2$, $\hspace{0.25cm} 2\pi$
    }
]
\addplot [mark=none, ultra thick, red] {cos(deg(x))};
\end{axis}
\end{tikzpicture}

\begin{tikzpicture}
\begin{axis}[
    Axis Style,
    xtick={-6.28318, -4.7123889, ..., 6.28318},
    scaled x ticks={real:3.1415},
    xtick scale label code/.code={},
]
\addplot [mark=none, thin, brown] {tan(deg(x))};
\end{axis}
\end{tikzpicture}


\end{document}

Answer 1

You've asked quite a few questions! It's generally best to ask one, focused question on this site ;)

But, here's an attempt to get through your questions....

Labels, such as $y = \sin(x), would make the graphs more presentable. I guess directly under the y-axis.

You could use, for example,

\begin{axis}[
    Axis Style,
    xtick={
        -6.28318, -4.7123889, -3.14159, -1.5708,
        1.5708, 3.14159, 4.7123889, 6.28318
    },
    xticklabels={
        $-2\pi$, $-3\pi/2$, $-\pi$, $\pi/2$,
        $\pi/2$, $\pi$, $3\pi/2$, $2\pi$
    },
    ylabel={$y=\sin(x)$}, %<---- new bit
]

or else, for example,

...
\addplot [mark=none, ultra thick, blue] {sin(deg(x))};
\addlegendentry{$y=\sin(x)$} %<---- new bit
...

You might also want to adjust the legend position using legend pos=...

Arrowheads should be at both ends of the axes. They are currently at one end.

You can use,

\pgfkeys{/pgfplots/Axis Style/.style={
    width=.3\textwidth, %height=5cm,
    axis x line=center, 
    axis y line=middle, 
    samples=100,
    ymin=-1.5, ymax=1.5,
    xmin=-7.0, xmax=7.0,
    domain=-2*pi:2*pi,
    axis line style=<->,  %<----- new bit
}}

or otherwise use pgfplotsset (as demonstrated below).

The scale is suitable for the graphs of the sine and cosine functions but not for the tangent function. Maybe have the y-axis include -10 and 10.

Try using, for example,

\begin{axis}[
    Axis Style,
    xtick={-6.28318, -4.7123889, ..., 6.28318},
    scaled x ticks={real:3.1415},
    xtick scale label code/.code={},
    ymin=-10,ymax=10, %<---- new bit
]

Changing the scale for this graph may have the labels along the axes drawn over by the graph of the tangent function. Is there a code for putting labels (in nodes) over other features of a graph?

I believe you want the axis on top key for this.

Currently, the y-axes of the three graphs are aligned. How can I have the x-axes are aligned?

Remove the blank lines between your figures - note that you'll need to adjust the text width to get them to fit nicely - consider the geometry package for this.

What does mark=none instruct TikZ to do? What other options are there for mark?

mark=none means that you only get the curve, and not circles (or other marks) at the sample points along the way. There are a lot of other options - see Section 4.7 of the pgfplots for details.

Here's a complete bit of code to play with that implements the things I have mentioned.

% arara: pdflatex
% !arara: indent: {overwrite: yes}
\documentclass{article}
\usepackage{pgfplots}
\usepackage[margin=1cm]{geometry}
\pgfplotsset{Axis Style/.style={
    width=.3\textwidth, %height=5cm,
    axis x line=center,
    axis y line=middle,
    samples=100,
    ymin=-1.5, ymax=1.5,
    xmin=-7.0, xmax=7.0,
    domain=-2*pi:2*pi,
    axis line style=<->,
}}


\begin{document}

\begin{tikzpicture}
    \begin{axis}[
            Axis Style,
            xtick={
                -6.28318, -4.7123889, -3.14159, -1.5708,
                1.5708, 3.14159, 4.7123889, 6.28318
            },
            xticklabels={
                $-2\pi$, $-3\pi/2$, $-\pi$, $\pi/2$,
                $\pi/2$, $\pi$, $3\pi/2$, $2\pi$
            },
            ylabel={$y=\sin(x)$},
        ]
        \addplot [mark=none, ultra thick, blue] {sin(deg(x))};
        \addlegendentry{$y=\sin(x)$}
    \end{axis}
\end{tikzpicture}
\begin{tikzpicture}
    \begin{axis}[
            Axis Style,
            xtick={
                -6.28318, -4.7123889, -3.14159, -1.5708,
                1.5708, 3.14159, 4.7123889, 6.28318
            },
            xticklabels={
                $-2\pi$, $-3\pi/2$, $-\pi\hspace{0.30cm}$, $\pi/2$,
                $\pi/2$, $\pi\hspace{0.10cm}$, $3\pi/2$, $\hspace{0.25cm} 2\pi$
            }
        ]
        \addplot [mark=none, ultra thick, red] {cos(deg(x))};
    \end{axis}
\end{tikzpicture}
\begin{tikzpicture}
    \begin{axis}[
            Axis Style,
            xtick={-6.28318, -4.7123889, ..., 6.28318},
            scaled x ticks={real:3.1415},
            xtick scale label code/.code={},
            ymin=-10,ymax=10,
        ]
        \addplot [mark=none, ultra thick, brown] {tan(deg(x))};
    \end{axis}
\end{tikzpicture}


\end{document}

Answer 2

An easy-to-customize template with PSTricks.

\documentclass[pstricks,border=0pt,12pt,dvipsnames]{standalone}
\usepackage{amsmath}
\usepackage{pstricks-add}
\usepackage{pst-plot}
% \usepackage{pst-math} provides ATAN
\usepackage[nomessages]{fp}

\FPeval\XMin{0-2*pi}
\FPeval\XMax{2*pi}
\FPeval\YMin{0-pi}
\FPeval\YMax{5*pi/3}

\FPeval\XOL{0-1/3} % of DeltaX
\FPeval\XOR{1/3} % of DeltaX
\FPeval\YOB{0-1/3} % of DeltaY
\FPeval\YOT{1/3} % of DeltaY

\FPset\TrigLabelBase{3}
\FPeval\DeltaX{pi/TrigLabelBase}
\FPeval\DeltaY{pi/TrigLabelBase}

\FPeval\AxisL{XMin+DeltaX*XOL}
\FPeval\AxisR{XMax+DeltaX*XOR}
\FPeval\AxisB{YMin+DeltaY*YOB}
\FPeval\AxisT{YMax+DeltaY*YOT}

\newlength\Width\Width=12cm
\newlength\Height\Height=8cm

\newlength\llx\llx=-5pt
\newlength\urx\urx=15pt
\newlength\lly\lly=-5pt
\newlength\ury\ury=15pt


\psset
{
    llx=\llx,
    lly=\lly,
    urx=\urx,
    ury=\ury,
    xtrigLabels=true,
    ytrigLabels=true,
    trigLabelBase=\TrigLabelBase,
    labelFontSize=\scriptstyle,
    xAxisLabel=$x$,
    yAxisLabel=$y$,
    algebraic,
    plotpoints=500,
}

\def\f{Pi*sin(x)}
\def\fp{Derive(1,\f)}

\def\g{2*asin(x/(2*Pi))}

\begin{document}
\pslegend[lt]{%
    \color{NavyBlue}\rule{12pt}{1pt} & \color{NavyBlue} $y=\pi\sin x$ \\
    \color{Red}\rule{12pt}{1pt} & \color{Red} $y=\pi\cos x$ \\
    \color{ForestGreen}\rule{12pt}{1pt} & \color{ForestGreen} $y=2\sin^{-1}\left(\tfrac{x}{2\pi}\right)$
}
\begin{psgraph}
    [
        dx=\DeltaX,
        dy=\DeltaY,
        linecolor=gray,
        tickcolor=gray,
        ticksize=-3pt 3pt,
        axespos=top,
    ]{<->}(0,0)(\AxisL,\AxisB)(\AxisR,\AxisT){\dimexpr\Width-\urx+\llx}{!}%{\dimexpr\Height-\ury+\lly}
    \psaxes
    [
        dx=\DeltaX,
        dy=\DeltaY,
        labels=none,
        subticks=5,
        tickwidth=.4pt,
        subtickwidth=.2pt,
        tickcolor=Red!30,
        subtickcolor=ForestGreen!30,
        xticksize=\YMin\space \YMax,
        yticksize=\XMin\space \XMax,
        subticksize=1,
    ](0,0)(\XMin,\YMin)(\XMax,\YMax)
    \psplot[linecolor=NavyBlue]{\XMin}{\XMax}{\f}
    \psplot[linecolor=Red]{\XMin}{\XMax}{\fp}
    \psplot[linecolor=ForestGreen]{\XMin}{\XMax}{\g}
    % -------------------
    % tangent and normal
    \psplotTangent[linecolor=Magenta]{Pi 3 div 2 mul}{1.5}{\f}
    \psplotTangent[linecolor=Cyan,Derive={-1/\fp}]{Pi 3 div 2 mul}{1.5}{\f}
    %
    % sum
    \psplot[linecolor=Rhodamine]{\XMin}{\XMax}{Sum(i,0,1,7,(-1)^(i)*x^(2*i)/fact(2*i))}
\end{psgraph}
\end{document}

Documentation

This section explains the settings used in the template.