10.5.2.2 Options Controlling Simplification

Option variable: %piargs ¶

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)   ]

Categories: Trigonometric functions · Simplification flags and variables ·

Option variable: %iargs ¶

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)

Categories: Trigonometric functions · Hyperbolic functions · Simplification flags and variables ·

Option variable: halfangles ¶

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)

Categories: Trigonometric functions · Simplification flags and variables ·

Option variable: trigsign ¶

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.

Categories: Trigonometric functions · Simplification flags and variables ·