Maxima: trigonometric numbers in radical form

Beginner's question for Maxima: how can I obtain trigonometric numbers in radical form?

For example, this expression evaluates nicely:

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

But this one does not:

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

I would expect it to show something like this:

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

See, for example, the output from Wolfram Alpha:

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

Solution

From the Maxima documentation for piargs, which is true by default:

When %piargs is true, trigonometric functions are simplified to algebraic constants when the argument is an integer multiple of %pi, %pi/2, %pi/3, %pi/4, or %pi/6.

From the Maxima documentation for ntrig:

The ntrig package contains a set of simplification rules that are used to simplify trigonometric function whose arguments are of the form f(n%pi/10) where f is any of the functions sin, cos, tan, csc, sec and cot.

This will work for 3π/5, but not for more complex values like 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, the trigeval function from the trigtools package will 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 is some documentation for trigtools, but because it is part of the 3rd-party contrib packages, it is not as well maintained. The source code for trigtools hasn't been updated since 2013.

Also, trigeval only seems to work for angles corresponding to regular polygons, and not for trigonometric numbers in general. For example, cos (π / 23) = -(1/2)(-1)^22/23(1+(-1)^2/23), but trigeval is unhelpful in this case:

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

Credit goes to Serge de Marre and Raymond Toy on the maxima-discuss mailing list, as well as David Billinghurst on the Maxima Area 51 Stackexchange.

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