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Default value: true
When %piargs is true,
trigonometric functions are simplified to algebraic constants
when the argument is an integer multiple
of 
\(\pi,\) 
\(\pi/2,\) 
\(\pi/4,\) or 
\(\pi/6.\)
Maxima knows some identities which can be applied when \(\pi\), etc., are multiplied by an integer variable (that is, a symbol declared to be integer).
Examples:
(%i1) %piargs : false$
(%i2) [sin (%pi), sin (%pi/2), sin (%pi/3)];
                                %pi       %pi
(%o2)            [sin(%pi), sin(---), sin(---)]
                                 2         3
(%i3) [sin (%pi/4), sin (%pi/5), sin (%pi/6)];
                      %pi       %pi       %pi
(%o3)            [sin(---), sin(---), sin(---)]
                       4         5         6
(%i4) %piargs : true$
(%i5) [sin (%pi), sin (%pi/2), sin (%pi/3)];
                                sqrt(3)
(%o5)                    [0, 1, -------]
                                   2
(%i6) [sin (%pi/4), sin (%pi/5), sin (%pi/6)];
                         1         %pi   1
(%o6)                [-------, sin(---), -]
                      sqrt(2)       5    2
(%i7) [cos (%pi/3), cos (10*%pi/3), tan (10*%pi/3),
       cos (sqrt(2)*%pi/3)];
                1    1               sqrt(2) %pi
(%o7)          [-, - -, sqrt(3), cos(-----------)]
                2    2                    3
Some identities are applied when \(\pi\) and \(\pi/2\) are multiplied by an integer variable.
(%i1) declare (n, integer, m, even)$
(%i2) [sin (%pi * n), cos (%pi * m), sin (%pi/2 * m),
       cos (%pi/2 * m)];
                                      m/2
(%o2)                  [0, 1, 0, (- 1)   ]
Default value: true
When %iargs is true,
trigonometric functions are simplified to hyperbolic functions
when the argument is apparently a multiple of the imaginary
unit 
\(i.\)
Even when the argument is demonstrably real, the simplification is applied; Maxima considers only whether the argument is a literal multiple of \(i.\)
Examples:
(%i1) %iargs : false$
(%i2) [sin (%i * x), cos (%i * x), tan (%i * x)]; (%o2) [sin(%i x), cos(%i x), tan(%i x)]
(%i3) %iargs : true$
(%i4) [sin (%i * x), cos (%i * x), tan (%i * x)]; (%o4) [%i sinh(x), cosh(x), %i tanh(x)]
Even when the argument is demonstrably real, the simplification is applied.
(%i1) declare (x, imaginary)$
(%i2) [featurep (x, imaginary), featurep (x, real)]; (%o2) [true, false]
(%i3) sin (%i * x); (%o3) %i sinh(x)
Default value: false
When halfangles is true, trigonometric functions of arguments 
expr/2 are simplified to functions of expr.
For a real argument x in the interval \(0 \le x < 2\pi,\) \(\sin{x\over 2}\) simplifies to a simple formula: $$ {\sqrt{1-\cos x}\over\sqrt{2}} $$
A complicated factor is needed to make this formula correct for all complex arguments z = x+iy: $$ (-1)^{\lfloor{x/(2\pi)}\rfloor} \left[1-\rm{unit\_step}(-y) \left(1+(-1)^{\lfloor{x/(2\pi)}\rfloor - \lceil{x/(2\pi)}\rceil}\right)\right] $$
Maxima knows this factor and similar factors for the functions sin, 
cos, sinh, and cosh.  For special values of the argument 
z these factors simplify accordingly.
Examples:
(%i1) halfangles : false$
(%i2) sin (x / 2);
                                 x
(%o2)                        sin(-)
                                 2
(%i3) halfangles : true$
(%i4) sin (x / 2);
                            x
                    floor(-----)
                          2 %pi
               (- 1)             sqrt(1 - cos(x))
(%o4)          ----------------------------------
                            sqrt(2)
(%i5) assume(x>0, x<2*%pi)$
(%i6) sin(x / 2);
                        sqrt(1 - cos(x))
(%o6)                   ----------------
                            sqrt(2)
Default value: true
When trigsign is true, it permits simplification of negative
arguments to trigonometric functions.  E.g., 
\(\sin(-x)\) will
become 
\(-\sin x\) only if trigsign is true.