symbolic math - Maxima: trigonometric numbers in radical form -


beginner's question maxima: how can obtain trigonometric numbers in radical form?

for example, expression evaluates nicely:

(%i) cos( 3 * %pi / 4);                                        1 (%o)                              - -------                                     sqrt(2)

but 1 not:

(%i) cos(3 * %pi / 5);                                      3 %pi (%o)                             cos(-----)                                        5

i expect show this:

(%i) cos( 3 * %pi / 5);                                   1 - sqrt(5) (%o)                              -----------                                        4

see, example, output wolfram alpha:

http://www.wolframalpha.com/input/?i=cos%283+pi+%2f+5%29

from maxima documentation piargs, true default:

when %piargs true, trigonometric functions simplified algebraic constants when argument integer multiple of %pi, %pi/2, %pi/3, %pi/4, or %pi/6.

from maxima documentation ntrig:

the ntrig package contains set of simplification rules used simplify trigonometric function arguments of form f(n%pi/10) f of functions sin, cos, tan, csc, sec , cot.

this work 3π/5, not more complex values π/96:

(%i) load(ntrig); (%o)        /usr/share/maxima/5.34.1/share/trigonometry/ntrig.mac (%i) cos(3*%pi/5);                                  1 - sqrt(5) (%o)                             -----------                                       4 (%i) sin(4*%pi/10);                              sqrt(sqrt(5) + 5) (%o)                         -----------------                                     3/2                                    2 (%i) sin(%pi/96);                                       %pi (%o)                              sin(---)                                       96

to evaluate more complex results, trigeval function trigtools package work:

(%i) load(trigtools); (%o)   /usr/share/maxima/5.34.1/share/contrib/trigtools/trigtools.mac (%i) trigeval(sin(4*%pi/10));                               sqrt(sqrt(5) + 5) (%o)                          -----------------                                      3/2                                     2 (%i) trigeval(sin(%pi/96));                    9/8                        3/2         5/4              sqrt(2    - sqrt(sqrt(sqrt(3) + 2    + 1) + 2   )) (%o)         --------------------------------------------------                                     17/16                                    2

there documentation trigtools, because part of 3rd-party contrib packages, not maintained. source code trigtools hasn't been updated since 2013.

also, trigeval seems work angles corresponding regular polygons, , not trigonometric numbers in general. example, cos (π / 23) = -(1/2)(-1)22/23(1+(-1)2/23), trigeval unhelpful in case:

(%i) trigeval(cos(%pi/23));                                       %pi (%o)                              cos(---)                                       23

credit goes serge de marre , raymond toy on maxima-discuss mailing list, david billinghurst on maxima area 51 stackexchange.

relevant links other mailing lists:


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