Artifact e67817cd3113708dbc50798f217b2dd19e5ea0a2b038c33a5ed535abf9434f9f:
- Executable file
r37/doc/manual2/trigsimp.tex
— part of check-in
[f2fda60abd]
at
2011-09-02 18:13:33
on branch master
— Some historical releases purely for archival purposes
git-svn-id: https://svn.code.sf.net/p/reduce-algebra/code/trunk/historical@1375 2bfe0521-f11c-4a00-b80e-6202646ff360 (user: arthurcnorman@users.sourceforge.net, size: 5166) [annotate] [blame] [check-ins using] [more...]
- Executable file
r38/doc/manual2/trigsimp.tex
— part of check-in
[f2fda60abd]
at
2011-09-02 18:13:33
on branch master
— Some historical releases purely for archival purposes
git-svn-id: https://svn.code.sf.net/p/reduce-algebra/code/trunk/historical@1375 2bfe0521-f11c-4a00-b80e-6202646ff360 (user: arthurcnorman@users.sourceforge.net, size: 5166) [annotate] [blame] [check-ins using]
\chapter[TRIGSIMP: Trigonometric simplification]% {TRIGSIMP: Simplification and factorisation of trigonometric and hyperbolic functions} \label{TRIGSIMP} \typeout{{TRIGSIMP: Simplification and factorisation of trigonometric and hyperbolic functions}} {\footnotesize \begin{center} Wolfram Koepf, Andreas Bernig and Herbert Melenk\\ Konrad--Zuse--Zentrum f\"ur Informationstechnik Berlin \\ Takustra\"se 7 \\ D--14195 Berlin--Dahlem, Germany \\[0.05in] e--mail: Koepf@zib.de \end{center} } \ttindex{TRIGSIMP} There are three procedures included in TRIGSIMP: trigsimp, trigfactorize and triggcd. The first is for finding simplifications of trigonometric or hyperbolic expressions with many options, the second for factorising them and the third for finding the greatest common divisor of two trigonometric or hyperbolic polynomials. \section{Simplifiying trigonometric expressions} As there is no normal form for trigonometric and hyperbolic functions, the same function can convert in many different directions, {\em e.g. } $\sin(2x) \leftrightarrow 2\sin(x)\cos(x)$. The user has the possibility to give several parameters to the procedure {\tt trigsimp} in order to influence the direction of transformations. The decision whether a rational expression in trigonometric and hyperbolic functions vanishes or not is possible. \ttindex{trigsimp} To simplify a function {\tt f}, one uses {\tt trigsimp(f[,options])}. Example: \begin{verbatim} 2: trigsimp(sin(x)^2+cos(x)^2); 1 \end{verbatim} Possible options are (* denotes the default): \begin{enumerate} \item {\tt sin} (*) or {\tt cos}\index{trigsimp ! sin}\index{trigsimp ! cos} \item {\tt sinh} (*) or {\tt cosh}\index{trigsimp ! sinh}\index{trigsimp ! cosh} \item {\tt expand} (*) or {\tt combine} or {\tt compact}\index{trigsimp ! expand}\index{trigsimp ! combine}\index{trigsimp ! compact} \item {\tt hyp} or {\tt trig} or {\tt expon}\index{trigsimp ! hyp}\index{trigsimp ! trig}\index{trigsimp ! expon} \item {\tt keepalltrig}\index{trigsimp ! keepalltrig} \end{enumerate} From each group one can use at most one option, otherwise an error message will occur. The first group fixes the preference used while transforming a trigonometric expression. The second group is the equivalent for the hyperbolic functions. The third group determines the type of transformations. With the default {\tt expand}, an expression is written in a form only using single arguments and no sums of arguments. With {\tt combine}, products of trigonometric functions are transformed to trigonometric functions involving sums of arguments. \begin{verbatim} trigsimp(sin(x)^2,cos); 2 - cos(x) + 1 trigsimp(sin(x)*cos(y),combine); sin(x - y) + sin(x + y) ------------------------- 2 \end{verbatim} With {\tt compact}, the \REDUCE\ operator {\tt compact} (see chapter~\ref{COMPACT}) is applied to {\tt f}. This leads often to a simple form, but in contrast to {\tt expand} one doesn't get a normal form. \begin{verbatim} trigsimp((1-sin(x)**2)**20*(1-cos(x)**2)**20,compact); 40 40 cos(x) *sin(x) \end{verbatim} With the fourth group each expression is transformed to a trigonometric, hyperbolic or exponential form: \begin{verbatim} trigsimp(sin(x),hyp); - sinh(i*x)*i trigsimp(e^x,trig); x x cos(---) + sin(---)*i i i \end{verbatim} Usually, {\tt tan}, {\tt cot}, {\tt sec}, {\tt csc} are expressed in terms of {\tt sin} and {\tt cos}. It can be sometimes useful to avoid this, which is handled by the option {\tt keepalltrig}: \begin{verbatim} trigsimp(tan(x+y),keepalltrig); - (tan(x) + tan(y)) ---------------------- tan(x)*tan(y) - 1 \end{verbatim} It is possible to use the options of different groups simultaneously. \section{Factorising trigonometric expressions} With {\tt trigfactorize(p,x)} one can factorise the trigonometric or hyperbolic polynomial {\tt p} with respect to the argument x. Example: \ttindex{trigfactorize} \begin{verbatim} trigfactorize(sin(x),x/2); x x {2,cos(---),sin(---)} 2 2 \end{verbatim} If the polynomial is not coordinated or balanced the output will equal the input. In this case, changing the value for x can help to find a factorisation: \begin{verbatim} trigfactorize(1+cos(x),x); {cos(x) + 1} trigfactorize(1+cos(x),x/2); x x {2,cos(---),cos(---)} 2 2 \end{verbatim} \section{GCDs of trigonometric expressions} The operator {\tt triggcd}\ttindex{triggcd} is an application of {\tt trigfactorize}. With its help the user can find the greatest common divisor of two trigonometric or hyperbolic polynomials. The syntax is: {\tt triggcd(p,q,x)}, where p and q are the polynomials and x is the smallest unit to use. Example: \begin{verbatim} triggcd(sin(x),1+cos(x),x/2); x cos(---) 2 triggcd(sin(x),1+cos(x),x); 1 \end{verbatim} See also the ASSIST package (chapter~\ref{ASSIST}).