programmatically drawing sinusoids with tikz

Question

Here's eight full oscillations of a sine wave:

Here's the code:

\documentclass{standalone}
\usepackage{tikz}

\begin{document}

    \begin{tikzpicture}
        \draw[gray!20] (0,-2) grid (32,2); %Grid
        \draw[black,-] (0, 0)  --  (32,0); %X-Axis
        \draw[black,-] (0,-2)  --  (0 ,2); %Y-Axis
        \draw[blue]    (0 ,0)              %Origin
        %Sine-Wave%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
            sin (1 ,2) cos (2 ,0) sin (3 ,-2) cos (4 ,0) sin (5 ,2) cos (6 ,0) sin (7 ,-2) cos (8 ,0)
            sin (9 ,2) cos (10,0) sin (11,-2) cos (12,0) sin (13,2) cos (14,0) sin (15,-2) cos (16,0) 
            sin (17,2) cos (18,0) sin (19,-2) cos (20,0) sin (21,2) cos (22,0) sin (23,-2) cos (24,0) 
            sin (25,2) cos (26,0) sin (27,-2) cos (28,0) sin (29,2) cos (30,0) sin (31,-2) cos (32,0);
        \end{tikzpicture}

\end{document}

My question isn't necessarily about sine waves or trigonometric functions. It's more about writing algorithms with LaTeX and using it like a real programming language.

Notice how I manually specified 32 pairs of coordinates and referred to both sin & cos 16 times just to draw 8 iterations of the same curve? Say, I want to generate a series of different standalone waves that are similar but different to this one. I don't want to have to type in every single coordinate every time, I want to automate all the boring, repetitive stuff.

Now, I don't need somebody to just draw it for me and dump the code, but rather share and explain the different tricks & techniques you can use to solve it in a programmatic way. A lot of us aren't aware of the tools at our disposal. The goal is to show the rest of us how to approach and solve these problems on our own so we can learn to do it for ourselves.

Answer 1

For drawing of your sinus function I would use plot macro:

\documentclass[tikz, margin=3mm]{standalone}

\begin{document}
    \begin{tikzpicture}
\draw[gray,thin] (0,-2) grid[xstep=pi/2,ystep=1] (8*pi,2); %Grid
\draw   (0, 0)  --  (8*pi,0)  %X-Axis
        (0,-2)  --  (0 ,2);   %Y-Axis
% Sine-Wave, \x r means to convert '\x' from degrees to radians
\draw[blue] plot[domain=0:16*pi, samples=320] (0.5*\x,{2*sin(\x r)});
    \end{tikzpicture}
\end{document}

Edit:

• TikZ provide macro plot for drawing of function. Its use is described in section 22.5 Plotting a Function, pp 339 in the TikZ & PGF Manual (version 3.1.4a).

• Since I define domain in radians, to argument of sin function is add r: sin(\x r), which make conversion of function argument from radians to degrees.

• in defining domain is used small trick. Since domain of eight function intervals is longer than text width, domain is shortened to half (to 16*pi`` and the same time the widths of intervals is shortened to0.5*\x` (this double the function frequency):

\draw[blue] plot[domain=0:16*pi, samples=320] (0.5*\x,{2*sin(\x r)});

• That drawn function is smooth, I used 20 samples per radian (all together 320 samples)

Answer 2

Giving frequency, period and amplitude of sine wave with color as options in pic set. The grid is drawn according to the frequency, period and amplitude values. foreach loop in tikzset is started with zero, and the frequency is drawn with 0 to f-1. Drawing the sinusoid is based on TiKz fundamentals.

\documentclass[tikz,border=10pt]{standalone}
\tikzset{%
pics/sw/.style args={color=#1,f=#2,p=#3,a=#4}{%
code={%
    \def\f{#2}%Frequency
    \def\p{#3}%Period
    \def\a{#4}%Amplitude
\draw[gray!20] (0,-\a) grid (\p*\f,\a); %Grid
    \draw[black,-] (0, 0)  --  (\p*\f,0); %X-Axis
    \draw[black,-] (0,-\a)  --  (0 ,\a); %Y-Axis
    \pgfmathsetmacro{\k}{#2-1}
\foreach \t in{0,1,...,\k}{
\draw[thick,#1] ({\t*\p},0) sin (#3/4+\t*\p,#4)cos(#3/2+\t*\p,0)sin(#3*3/4+\t*\p,-#4)cos(#3+\t*\p,0);%
}%
}
 }
 }  
   \begin{document}
    \begin{tikzpicture}
\pic {sw={color=blue,f=8,p=4,a=2}};
    \end{tikzpicture}

\end{document}

For f=8, p=4 and a=2: \pic {sw={color=blue,f=8,p=4,a=2}};

For f=4, p=5 and a=3: \pic {sw={color=red,f=4,p=5,a=3}};

ADDENDUM: Adding some features to code:

\documentclass[tikz,border=10pt]{standalone}
\tikzset{%
pics/sw/.style args={color=#1,f=#2,p=#3,a=#4}{%
code={%
    \def\f{#2}%Frequency
    \def\p{#3}%Period
    \def\a{#4}%Amplitude
\draw[gray!50] (0,-\a) grid (\p*\f,\a); %Grid
    \pgfmathsetmacro{\n}{#2*#3}
\foreach \x in{0,1,...,\n}{
\node at (\x,0)[below right]{\x};}
    \draw[black,->] (0, 0)  --  (\p*\f+1,0)node[above]{$x$}; %X-Axis
    \draw[black,->] (0,-\a)node[left]{-$#4$}  --  (0 ,\a+1)node[right]{$y$}; %Y-Axis
    \node at (0,\a)[left]{$#4$};
    \pgfmathsetmacro{\k}{#2-1}
\foreach \t in{0,1,...,\k}{
\draw[ultra thick,#1] ({\t*\p},0) sin (#3/4+\t*\p,#4)cos(#3/2+\t*\p,0)sin(#3*3/4+\t*\p,-#4)cos(#3+\t*\p,0);%
}%
\node at (\n/2,\a+1){$y=\sin\,x$};%
}
 }
 }  
   \begin{document}
    \begin{tikzpicture}
\pic {sw={color=cyan,f=6,p=4,a=3}};
    \end{tikzpicture}

\end{document}

Answer 3

A very crude starting point with foreach loops. Not elegant (IMO), but gives one possibility.

\documentclass{standalone}
\usepackage{tikz}

\begin{document}
    \begin{tikzpicture}
    \draw[gray!20] (0,-2) grid (32,2); %Grid
    \draw[black,-] (0, 0)  --  (32,0); %X-Axis
    \draw[black,-] (0,-2)  --  (0 ,2); %Y-Axis
    %% incorporating @marmot's (user121799) suggestion      
    \draw[blue] foreach \x in {1,5,...,29} { ({\x-1} ,0) sin ({\x} ,2) cos ({\x+1} ,0) sin ({\x+2}, -2) cos ({\x+3} ,0)};
    \end{tikzpicture}
\end{document}

Nevertheless, I am pretty sure, you can do much more than this :)