/* Original version of this file copyright 1999 by Michael Wester,
 * and retrieved from http://www.math.unm.edu/~wester/demos/Trigonometry/problems.macsyma
 * circa 2006-10-23.
 *
 * Released under the terms of the GNU General Public License, version 2,
 * per message dated 2007-06-03 from Michael Wester to Robert Dodier
 * (contained in the file wester-gpl-permission-message.txt).
 *
 * See: "A Critique of the Mathematical Abilities of CA Systems"
 * by Michael Wester, pp 25--60 in
 * "Computer Algebra Systems: A Practical Guide", edited by Michael J. Wester
 * and published by John Wiley and Sons, Chichester, United Kingdom, 1999.
 */
/* ----------[ M a c s y m a ]---------- */
/* ---------- Initialization ---------- */
showtime: all$
prederror: false$
/* ---------- Trigonometry ---------- */
/* => - [(sqrt(5) + 1) sqrt(2)]/[(sqrt(5) - 1) sqrt(sqrt(5) + 5)]
      = - sqrt[1 + 2/sqrt(5)]
   From B. F. Caviness, Robert P. Gilbert, Wolfram Koepf, Roman Shtokhamer and
   David W. Wood, _An Introduction to Applied Symbolic Computation using
   MACSYMA_, University of Delaware, draft of December 14, 1993, section 2.3.3.
   */
tan(7*%pi/10);
ev(%, %piargs:all);
factor(rootscontract(%));
/* => - cos 3 */
sqrt((1 + cos(6))/2);
/* cos(n pi) + sin((4 n - 1)/2 pi) => (-1)^n - 1 for integer n */
declare(n, integer)$
cos(n*%pi) + sin((4*n - 1)/2 * %pi);
/* cos(cos(n pi) pi) + sin(cos(n pi) pi/2) => -1 + (-1)^n for integer n */
cos(cos(n*%pi)*%pi) + sin(cos(n*%pi)*%pi/2);
/* sin([n^5/5 + n^4/2 + n^3/3 - n/30] pi) => 0 for integer n
   [Paul Zimmermann] */
sin((n^5/5 + n^4/2 + n^3/3 - n/30) * %pi);
remove(n, integer)$
/* | cos x |, | sin x | => - cos x, - sin x  for  - 3 pi < x < - 5/2 pi */
assume(-3*%pi < x, x < -5/2*%pi)$
[abs(cos(x)), abs(sin(x))];
forget(-3*%pi < x, x < -5/2*%pi)$
/* Trigonometric manipulations---these are typically difficult for students */
r: cos(3*x)/cos(x);
/*  => cos(x)^2 - 3 sin(x)^2 or similar */
ratsimp(trigexpand(r));
/* => 2 cos(2 x) - 1 */
poissimp(ratsimp(trigexpand(r)));
ratsimp(trigreduce(ratsimp(trigexpand(r))));
/* Use rewrite rules => cos(x)^2 - 3 sin(x)^2 */
matchdeclare(n, integerp, x, true)$
defrule(cos_angles, cos(n*x),
        cos((n - 1)*x) * cos(x) - sin((n - 1)*x) * sin(x));
defrule(sin_angles, sin(n*x),
        sin((n - 1)*x) * cos(x) + cos((n - 1)*x) * sin(x));
apply2(r, cos_angles, sin_angles);
ratsimp(%);
remvalue(r)$
/* Here is a tricky way of writing 0/0 */
expr: (tan(x)^2 + 1 - sec(x)^2)/(sin(x)^2 + cos(x)^2 - 1);
/* Let's try simplifying this expression! */
trigsimp(expr);
/* What is its limit at zero? */
limit(expr, x, 0);
/* What is the derivative? */
diff(expr, x);
remvalue(expr)$
