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\documentstyle[11pt,reduce]{article} \title{{\tt TRIGSIMP}\\ A REDUCE Package for the Simplification and Factorization of Trigonometric and Hyperbolic Functions} \date{} \author{Wolfram Koepf\\ Andreas Bernig\\ Herbert Melenk\\ ZIB Berlin \\ email: {\tt Koepf@ZIB-Berlin.de}} \begin{document} \maketitle \section{Introduction} The REDUCE package TRIGSIMP is a useful tool for all kinds of trigonometric and hyperbolic simplification and factorization. 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 factorizing them and the third for finding the greatest common divisor of two trigonometric or hyperbolic polynomials. To start the package it must be loaded by: {\small \begin{verbatim} 1: load trigsimp; \end{verbatim} }\noindent \section{\REDUCE{} operator {\tt trigsimp}} As there is no normal form for trigonometric and hyperbolic functions, the same function can convert in many different directions, 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. To simplify a function {\tt f}, one uses {\tt trigsimp(f[,options])}. Example: {\small \begin{verbatim} 2: trigsimp(sin(x)^2+cos(x)^2); 1 \end{verbatim} }\noindent Possible options are (* denotes the default): \begin{enumerate} \item {\tt sin} (*) or {\tt cos} \item {\tt sinh} (*) or {\tt cosh} \item {\tt expand} (*) or {\tt combine} or {\tt compact} \item {\tt hyp} or {\tt trig} or {\tt expon} \item {\tt 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: {\small \begin{verbatim} 3: trigsimp(sin(x)^2); 2 sin(x) 4: trigsimp(sin(x)^2,cos); 2 - cos(x) + 1 \end{verbatim} }\noindent 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: {\small \begin{verbatim} 5: trigsimp(sin(2x+y)); 2 2*cos(x)*cos(y)*sin(x) - 2*sin(x) *sin(y) + sin(y) \end{verbatim} }\noindent With {\tt combine}, products of trigonometric functions are transformed to trigonometric functions involving sums of arguments: {\small \begin{verbatim} 6: trigsimp(sin(x)*cos(y),combine); sin(x - y) + sin(x + y) ------------------------- 2 \end{verbatim} }\noindent With {\tt compact}, the REDUCE operator {\tt compact} \cite{hearns} 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. Example for {\tt compact}: {\small \begin{verbatim} 7: trigsimp((1-sin(x)**2)**20*(1-cos(x)**2)**20,compact); 40 40 cos(x) *sin(x) \end{verbatim} }\noindent With the fourth group each expression is transformed to a trigonometric, hyperbolic or exponential form: {\small \begin{verbatim} 8: trigsimp(sin(x),hyp); - sinh(i*x)*i 9: trigsimp(sinh(x),expon); 2*x e - 1 ---------- x 2*e 10: trigsimp(e^x,trig); x x cos(---) + sin(---)*i i i \end{verbatim} }\noindent 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}: {\small \begin{verbatim} 11: trigsimp(tan(x+y),keepalltrig); - (tan(x) + tan(y)) ---------------------- tan(x)*tan(y) - 1 \end{verbatim} }\noindent It is possible to use the options of different groups simultaneously: {\small \begin{verbatim} 12: trigsimp(sin(x)**4,cos,combine); cos(4*x) - 4*cos(2*x) + 3 --------------------------- 8 \end{verbatim} }\noindent Sometimes, it is necessary to handle an expression in different steps: {\small \begin{verbatim} 13: trigsimp((sinh(x)+cosh(x))**n+(cosh(x)-sinh(x))**n,expon); 2*n*x e + 1 ------------ n*x e 14: trigsimp(ws,hyp); 2*cosh(n*x) 15: trigsimp((cosh(a*n)*sinh(a)*sinh(p)+cosh(a)*sinh(a*n)*sinh(p)+ sinh(a - p)*sinh(a*n))/sinh(a)); cosh(a*n)*sinh(p) + cosh(p)*sinh(a*n) 16: trigsimp(ws,combine); sinh(a*n + p) \end{verbatim} }\noindent \section{\REDUCE{} operator {\tt trigfactorize}} With {\tt trigfactorize(p,x)} one can factorize the trigonometric or hyperbolic polynomial {\tt p} with respect to the argument x. Example: {\small \begin{verbatim} 17: trigfactorize(sin(x),x/2); x x {2,cos(---),sin(---)} 2 2 \end{verbatim} }\noindent If the polynomial is not coordinated or balanced \cite{art}, the output will equal the input. In this case, changing the value for x can help to find a factorization: {\small \begin{verbatim} 18: trigfactorize(1+cos(x),x); {cos(x) + 1} 19: trigfactorize(1+cos(x),x/2); x x {2,cos(---),cos(---)} 2 2 \end{verbatim} }\noindent The polynomial can consist of both trigonometric and hyperbolic functions: {\small \begin{verbatim} 20: trigfactorize(sin(2x)*sinh(2x),x); {4, cos(x), sin(x), cosh(x), sinh(x)} \end{verbatim} }\noindent \section{\REDUCE{} operator {\tt triggcd}} The operator {\tt triggcd} is an application of {\tt trigfactorize}. With its help the user can find the greatest common divisor of two trigonometric or hyperbolic polynomials. It uses the method described in \cite{art}. The syntax is: {\tt triggcd(p,q,x)}, where p and q are the polynomials and x is the smallest unit to use. Example: {\small \begin{verbatim} 21: triggcd(sin(x),1+cos(x),x/2); x cos(---) 2 22: triggcd(sin(x),1+cos(x),x); 1 \end{verbatim} }\noindent The polynomials p and q can consist of both trigonometric and hyperbolic functions: {\small \begin{verbatim} 23: triggcd(sin(2x)*sinh(2x),(1-cos(2x))*(1+cosh(2x)),x); cosh(x)*sin(x) \end{verbatim} }\noindent \section{Further Examples} With the help of the package the user can create identities: {\small \begin{verbatim} 24: trigsimp(tan(x)*tan(y)); sin(x)*sin(y) --------------- cos(x)*cos(y) 25: trigsimp(ws,combine); cos(x - y) - cos(x + y) ------------------------- cos(x - y) + cos(x + y) 26: trigsimp((sin(x-a)+sin(x+a))/(cos(x-a)+cos(x+a))); sin(x) -------- cos(x) 27: trigsimp(cosh(n*acosh(x))-cos(n*acos(x)),trig); 0 28: trigsimp(sec(a-b),keepalltrig); csc(a)*csc(b)*sec(a)*sec(b) ------------------------------- csc(a)*csc(b) + sec(a)*sec(b) 29: trigsimp(tan(a+b),keepalltrig); - (tan(a) + tan(b)) ---------------------- tan(a)*tan(b) - 1 30: trigsimp(ws,keepalltrig,combine); tan(a + b) \end{verbatim} }\noindent Some difficult expressions can be simplified: {\small \begin{verbatim} 31: df(sqrt(1+cos(x)),x,4); 4 2 2 2 (sqrt(cos(x) + 1)*( - 4*cos(x) - 20*cos(x) *sin(x) + 12*cos(x) 2 4 2 - 4*cos(x)*sin(x) + 8*cos(x) - 15*sin(x) + 16*sin(x) ))/(16 4 3 2 *(cos(x) + 4*cos(x) + 6*cos(x) + 4*cos(x) + 1)) 32: trigsimp(ws); sqrt(cos(x) + 1) ------------------ 16 33: load taylor; 34: taylor(sin(x+a)*cos(x+b),x,0,4); cos(b)*sin(a) + (cos(a)*cos(b) - sin(a)*sin(b))*x 2 - (cos(a)*sin(b) + cos(b)*sin(a))*x 2*( - cos(a)*cos(b) + sin(a)*sin(b)) 3 + --------------------------------------*x 3 cos(a)*sin(b) + cos(b)*sin(a) 4 5 + -------------------------------*x + O(x ) 3 35: trigsimp(ws,combine); sin(a - b) + sin(a + b) 2 2*cos(a + b) 3 ------------------------- + cos(a + b)*x - sin(a + b)*x - --------------*x 2 3 sin(a + b) 4 5 + ------------*x + O(x ) 3 \end{verbatim} }\noindent Certain integrals whose calculation was not possible in REDUCE (without preprocessing), are now computable: {\small \begin{verbatim} 36: int(trigsimp(sin(x+y)*cos(x-y)*tan(x)),x); 2 2 cos(x) *x - cos(x)*sin(x) - 2*cos(y)*log(cos(x))*sin(y) + sin(x) *x --------------------------------------------------------------------- 2 37: int(trigsimp(sin(x+y)*cos(x-y)/tan(x)),x); x 2 (cos(x)*sin(x) - 2*cos(y)*log(tan(---) + 1)*sin(y) 2 x + 2*cos(y)*log(tan(---))*sin(y) + x)/2 2 \end{verbatim} }\noindent Without the package, the integration fails, in the second case one doesn't receive an answer for many hours. {\small \begin{verbatim} 38: trigfactorize(sin(2x)*cos(y)**2,y/2); {2*cos(x)*sin(x), y y cos(---) + sin(---), 2 2 y y cos(---) + sin(---), 2 2 y y cos(---) - sin(---), 2 2 y y cos(---) - sin(---)} 2 2 39: trigfactorize(sin(y)**4-x**2,y); 2 2 { - sin(y) + x, - (sin(y) + x)} 40: trigfactorize(sin(x)*sinh(x),x/2); x x x x {4,cos(---),sin(---),cosh(---),sinh(---)} 2 2 2 2 41: triggcd(-5+cos(2x)-6sin(x),-7+cos(2x)-8sin(x),x/2); x x 2*cos(---)*sin(---) + 1 2 2 42: triggcd(1-2cosh(x)+cosh(2x),1+2cosh(x)+cosh(2x),x/2); x 2 2*sinh(---) + 1 2 \end{verbatim} } \begin{thebibliography}{99} \bibitem{art} Roach, Kelly: Difficulties with Trigonometrics. Notes of a talk. \bibitem{hearns} Hearn, A.C.: COMPACT User Manual. \end{thebibliography} \end{document}