% \iffalse
%% File: trig.dtx Copyright (C) 1993-1994 David Carlisle
%
%<*dtx>
          \ProvidesFile{trig.dtx}
%</dtx>
%<*!plain>
%<package&!plain>\NeedsTeXFormat{LaTeX2e}
%<package&!plain>\ProvidesPackage{trig}
%<driver>        \ProvidesFile{trig.drv}
% \fi
%                \ProvidesFile{trig.dtx}
                 [1994/10/16 v1.08 sin cos tan (DPC)]
%
% \iffalse
%</!plain>
%<*driver>
\documentclass{ltxdoc}
\usepackage{trig}
\begin{document}
 \DocInput{trig.dtx}
\end{document}
%</driver>
% \fi
%
% \GetFileInfo{trig.dtx}
% \title{The \textsf{trig} package\thanks{This file
%        has version number \fileversion, last
%        revised \filedate.}}
% \author{David Carlisle}
% \date{\filedate}
% \maketitle
%
% \CheckSum{246}
%
% \changes{v1.00}{1993/00/00}{Undocumented versions}
% \changes{v1.05}{1993/10/07}{Documented, added tan}
% \changes{v1.06}{1994/02/01}{Update for LaTeX2e}
% \changes{v1.07}{1994/03/15}{Use ltxdoc}
% \changes{v1.08}{1994/10/16}{Change \cs{@xc} to \cs{nin@ty}}
%
% \section{Introduction}
%
% These macros implement the trigonometric functions, sin, cos and tan.
% In each case two commands are defined. For instance the command
% |\CalculateSin{33}| may be isued at some point, and then anywhere
% later in the document, the command |\UseSin{33}| will return the
% decimal expansion of $\sin(33^\circ)$.
%
% The arguments to these macros do not have to be whole numbers,
% although in the case of whole numbers, \LaTeX\ or plain \TeX\ counters
% may be used. In \TeX{}Book syntax, arguments must be  of type:
% \meta{optional signs}\meta{factor}
%
% Some other examples are:\\
% |\CalculateSin{22.5}|, |\UseTan{\value{mycounter}}|,
% |\UseCos{\count@}|.
%
% Note that unlike the psfig macros, these save all previously
% computed values. This could easily be changed, but I thought that in
% many applications one would want many instances of the
% same value. (eg rotating all the headings of a table by the
% \emph{same} amount).
%
% I don't really like this need to pre-calculate the values, I
% originally implemented |\UseSin| so that it automatically calculated
% the value if it was not pre-stored. This worked fine in testing, until
% I remembered why one needs these values. You want to be able to say
% |\dimen2=\UseSin{30}\dimen0|. Which means that |\UseSin| must
% \emph{expand} to a \meta{factor}.
%
% \StopEventually{}
%
% \section{The Macros}
%
%    \begin{macrocode}
%<*package>
%    \end{macrocode}
%
% \begin{macro}{\nin@ty}\begin{macro}{\@clxx}
% \begin{macro}{\@lxxi}\begin{macro}{\@mmmmlxviii}
% Some useful constants for converting between degrees and radians.
% $$\frac{\pi}{180}\simeq\frac{355}{113\times180}=\frac{71}{4068}$$
%    \begin{macrocode}
\chardef\nin@ty=90
\chardef\@clxx=180
\chardef\@lxxi=71
\mathchardef\@mmmmlxviii=4068
%    \end{macrocode}
% \end{macro}\end{macro}\end{macro}\end{macro}
%
% The approximation to $\sin$. I experimented with various
% approximations based on Tchebicheff polynomials, and also some
% approximations from a SIAM handbook `Computer Approximations' However
% the standard Taylor series seems sufficiently accurate, and used by
% far the fewest \TeX\ tokens, as the coefficients are all rational.
% \begin{eqnarray*}
%  \sin(x)& \simeq& x - (1/3!)x^3 + (1/5!)x^5 - (1/7!)x^7 + (1/9!)x^9\\
%  &\simeq&\frac{((((7!/9!x^2-7!/7!)x^2+7!/5!)x^2 +7!/3!)x^2+7!/1!)x}
% {7!}\\
%  &=&\frac{((((1/72x^2-1)x^2+42)x^2 +840)x^2+5040)x}
% {5040}
% \end{eqnarray*}
% The nested form used above reduces the number of operations required.
% In order to further reduce the number of operations, and more
% importantly reduce the number of tokens used, we can precompute the
% coefficients. Note that we can not use $9!$ as the denominator as
% this would cause overflow of \TeX's arithmetic.
% \begin{macro}{\@coeffz}\begin{macro}{\@coeffa}\begin{macro}{\@coeffb}
% \begin{macro}{\@coeffc}\begin{macro}{\@coeffd}
% Save the coefficients as |\|(|math|)|char|s.
%    \begin{macrocode}
\chardef\@coeffz=72
%\chardef\@coefa=1
\chardef\@coefb=42
\mathchardef\@coefc=840
\mathchardef\@coefd=5040
%    \end{macrocode}
% \end{macro}\end{macro}\end{macro}\end{macro}\end{macro}
%
% \begin{macro}{\TG@rem@pt}
% The standard trick of getting a real number out of a \meta{dimen}.
% This gives a maximum accuracy of approx.\ 5 decimal places, which
% should be sufficient. It puts a space after the number, perhaps it
% shouldn't.
%    \begin{macrocode}
{\catcode`t=12\catcode`p=12\gdef\noPT#1pt{#1}}
\def\TG@rem@pt#1{\expandafter\noPT\the#1\space}
%    \end{macrocode}
% \end{macro}
%
% \begin{macro}{\TG@term}
% Compute one term of the above nested series. Multiply the previous sum
% by $x^2$ (stored in |\@tempb|, then add the next coefficient, |#1|.
%    \begin{macrocode}
\def\TG@term#1{%
 \dimen@\@tempb\dimen@
 \advance\dimen@ #1\p@}
%    \end{macrocode}
% \end{macro}
%
% \begin{macro}{\TG@series}
% Compute the above series. the value in degrees will be in |\dimen@|
% before this is called.
%    \begin{macrocode}
\def\TG@series{%
 \dimen@\@lxxi\dimen@
 \divide \dimen@ \@mmmmlxviii
%    \end{macrocode}
% |\dimen@| now contains the angle in radians, as a \meta{dimen}. We
% need to remove the units, so store the same value as a \meta{factor}
% in |\@tempa|.
%    \begin{macrocode}
 \edef\@tempa{\TG@rem@pt\dimen@}%
%    \end{macrocode}
% Now put $x^2$ in |\dimen@| and |\@tempb|.
%    \begin{macrocode}
 \dimen@\@tempa\dimen@
 \edef\@tempb{\TG@rem@pt\dimen@}%
%    \end{macrocode}
% The first coefficient is  $1/72$.
%    \begin{macrocode}
 \divide\dimen@\@coeffz
 \advance\dimen@\m@ne\p@
 \TG@term\@coefb
 \TG@term{-\@coefc}%
 \TG@term\@coefd
%    \end{macrocode}
% Now the cubic in $x^2$ is completed, so we need to multiply by $x$ and
% divide by $7!$.
%    \begin{macrocode}
 \dimen@\@tempa\dimen@
 \divide\dimen@ \@coefd}
%    \end{macrocode}
% \end{macro}
%
% \begin{macro}{\CalculateSin}
% If this angle has already been computed, do nothing, else store the
% angle, and call |\TG@@sin|.
%    \begin{macrocode}
\def\CalculateSin#1{{%
  \expandafter\ifx\csname sin(\number#1)\endcsname\relax
    \dimen@=#1\p@\TG@@sin
    \expandafter\xdef\csname sin(\number#1)\endcsname
                                    {\TG@rem@pt\dimen@}%
  \fi}}
%    \end{macrocode}
% \end{macro}
%
% \begin{macro}{\CalculateCos}
% As above, but use the relation $\cos(x) = \sin(90-x)$.
%    \begin{macrocode}
\def\CalculateCos#1{{%
  \expandafter\ifx\csname cos(\number#1)\endcsname\relax
    \dimen@=\nin@ty\p@
    \advance\dimen@-#1\p@
    \TG@@sin
    \expandafter\xdef\csname cos(\number#1)\endcsname
                                     {\TG@rem@pt\dimen@}%
  \fi}}
%    \end{macrocode}
% \end{macro}
%
% \begin{macro}{\TG@reduce}
% Repeatedly use one of the the relatations
% $\sin(x)=\sin(180-x)=\sin(-180-x)$ to get $x$ in the range $-90 \leq
% x\leq 90$. Then call |\TG@series|.
%    \begin{macrocode}
\def\TG@reduce#1#2{%
\dimen@#1#2\nin@ty\p@
  \advance\dimen@#2-\@clxx\p@
  \dimen@-\dimen@
  \TG@@sin}
%    \end{macrocode}
% \end{macro}
%
% \begin{macro}{\TG@@sin}
% Slightly cryptic, but it seems to  work\ldots
%    \begin{macrocode}
\def\TG@@sin{%
  \ifdim\TG@reduce>+%
  \else\ifdim\TG@reduce<-%
  \else\TG@series\fi\fi}%
%    \end{macrocode}
% \end{macro}
%
% \begin{macro}{\UseSin}
% \begin{macro}{\UseCos}
% Use a pre-computed value.
%    \begin{macrocode}
\def\UseSin#1{\csname sin(\number#1)\endcsname}
\def\UseCos#1{\csname cos(\number#1)\endcsname}
%    \end{macrocode}
% \end{macro}
% \end{macro}
%
% A few shortcuts to save space.
%    \begin{macrocode}
\chardef\z@num\z@
\expandafter\let\csname sin(0)\endcsname\z@num
\expandafter\let\csname cos(0)\endcsname\@ne
\expandafter\let\csname sin(90)\endcsname\@ne
\expandafter\let\csname cos(90)\endcsname\z@num
\expandafter\let\csname sin(-90)\endcsname\m@ne
\expandafter\let\csname cos(-90)\endcsname\z@num
\expandafter\let\csname sin(180)\endcsname\z@num
\expandafter\let\csname cos(180)\endcsname\m@ne
%    \end{macrocode}
%
% \begin{macro}{\CalculateTan}
% Originally I coded the Taylor series for tan, but it seems to be
% more accurate to just take the ratio of the sine and cosine.
% This is accurate to 4 decimal places for angles up to
% $50^\circ$, after that the accuracy tails off, giving
% 57.47894 instead of 57.2900 for $89^\circ$.
%    \begin{macrocode}
\def\CalculateTan#1{{%
  \expandafter\ifx\csname tan(\number#1)\endcsname\relax
    \CalculateSin{#1}%
    \CalculateCos{#1}%
    \@tempdima\UseCos{#1}\p@
    \divide\@tempdima\@iv
    \@tempdimb\UseSin{#1}\p@
    \@tempdimb\two@fourteen\@tempdimb
    \divide\@tempdimb\@tempdima
    \expandafter\xdef\csname tan(\number#1)\endcsname
                                        {\TG@rem@pt\@tempdimb}%
  \fi}}
%    \end{macrocode}
% \end{macro}
%
% \begin{macro}{\UseTan}
% Just like |\UseSin|.
%    \begin{macrocode}
\def\UseTan#1{\csname tan(\number#1)\endcsname}
%    \end{macrocode}
% \end{macro}
%
% \begin{macro}{\two@fourteen}
% \begin{macro}{\@iv}
% two constants needed to keep the division within \TeX's range.
%    \begin{macrocode}
\mathchardef\two@fourteen=16384
\chardef\@iv=4
%    \end{macrocode}
% \end{macro}
% \end{macro}
%
% Predefine $\tan(\pm90)$ to be an error.
%    \begin{macrocode}
\expandafter\def\csname tan(90)\endcsname{\errmessage{Infinite tan !}}
\expandafter\let\csname tan(-90)\expandafter\endcsname
                                       \csname tan(90)\endcsname
%    \end{macrocode}
%
%    \begin{macrocode}
%</package>
%    \end{macrocode}
%
% \Finale
%
\endinput
