Previous: , Up: Trigonometric Functions   [Contents][Index]

10.5.2 Functions and Variables for Trigonometric

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 \(%pi\), \(%pi/2\), \(%pi/3\), \(%pi/4\), 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)   ]
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)
Function: acos (x)

– Arc Cosine.

Categories: Trigonometric functions ·
Function: acosh (x)

– Hyperbolic Arc Cosine.

Categories: Hyperbolic functions ·
Function: acot (x)

– Arc Cotangent.

Categories: Trigonometric functions ·
Function: acoth (x)

– Hyperbolic Arc Cotangent.

Categories: Hyperbolic functions ·
Function: acsc (x)

– Arc Cosecant.

Categories: Trigonometric functions ·
Function: acsch (x)

– Hyperbolic Arc Cosecant.

Categories: Hyperbolic functions ·
Function: asec (x)

– Arc Secant.

Categories: Trigonometric functions ·
Function: asech (x)

– Hyperbolic Arc Secant.

Categories: Hyperbolic functions ·
Function: asin (x)

– Arc Sine.

Categories: Trigonometric functions ·
Function: asinh (x)

– Hyperbolic Arc Sine.

Categories: Hyperbolic functions ·
Function: atan (x)

– Arc Tangent.

See also atan2.

Categories: Trigonometric functions ·
Function: atan2 (y, x)

– yields the value of atan(y/x) in the interval -%pi to %pi.

See also atan.

Categories: Trigonometric functions ·
Function: atanh (x)

– Hyperbolic Arc Tangent.

Categories: Hyperbolic functions ·
Package: atrig1

The atrig1 package contains several additional simplification rules for inverse trigonometric functions. Together with rules already known to Maxima, the following angles are fully implemented: 0, %pi/6, %pi/4, %pi/3, and %pi/2. Corresponding angles in the other three quadrants are also available. Do load("atrig1"); to use them.

Function: cos (x)

– Cosine.

Categories: Trigonometric functions ·
Function: cosh (x)

– Hyperbolic Cosine.

Categories: Hyperbolic functions ·
Function: cot (x)

– Cotangent.

Categories: Trigonometric functions ·
Function: coth (x)

– Hyperbolic Cotangent.

Categories: Hyperbolic functions ·
Function: csc (x)

– Cosecant.

Categories: Trigonometric functions ·
Function: csch (x)

– Hyperbolic Cosecant.

Categories: Hyperbolic functions ·
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 < x < 2*%pi the sine of the half-angle simplifies to a simple formula:

                         sqrt(1 - cos(x))
                         ----------------
                             sqrt(2)

A complicated factor is needed to make this formula correct for all complex arguments z:

           realpart(z)
     floor(-----------)
              2 %pi
(- 1)                   (1 - unit_step(- imagpart(z))

                            realpart(z)            realpart(z)
                      floor(-----------) - ceiling(-----------)
                               2 %pi                  2 %pi
                ((- 1)                                          + 1))

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)
Package: 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.

Function: sec (x)

– Secant.

Categories: Trigonometric functions ·
Function: sech (x)

– Hyperbolic Secant.

Categories: Hyperbolic functions ·
Function: sin (x)

– Sine.

Categories: Trigonometric functions ·
Function: sinh (x)

– Hyperbolic Sine.

Categories: Hyperbolic functions ·
Function: tan (x)

– Tangent.

Categories: Trigonometric functions ·
Function: tanh (x)

– Hyperbolic Tangent.

Categories: Hyperbolic functions ·
Function: trigexpand (expr)

Expands trigonometric and hyperbolic functions of sums of angles and of multiple angles occurring in expr. For best results, expr should be expanded. To enhance user control of simplification, this function expands only one level at a time, expanding sums of angles or multiple angles. To obtain full expansion into sines and cosines immediately, set the switch trigexpand: true.

trigexpand is governed by the following global flags:

trigexpand

If true causes expansion of all expressions containing sin’s and cos’s occurring subsequently.

halfangles

If true causes half-angles to be simplified away.

trigexpandplus

Controls the "sum" rule for trigexpand, expansion of sums (e.g. sin(x + y)) will take place only if trigexpandplus is true.

trigexpandtimes

Controls the "product" rule for trigexpand, expansion of products (e.g. sin(2 x)) will take place only if trigexpandtimes is true.

Examples:

(%i1) x+sin(3*x)/sin(x),trigexpand=true,expand;
                         2            2
(%o1)              (- sin (x)) + 3 cos (x) + x
(%i2) trigexpand(sin(10*x+y));
(%o2)          cos(10 x) sin(y) + sin(10 x) cos(y)
Option variable: trigexpandplus

Default value: true

trigexpandplus controls the "sum" rule for trigexpand. Thus, when the trigexpand command is used or the trigexpand switch set to true, expansion of sums (e.g. sin(x+y)) will take place only if trigexpandplus is true.

Option variable: trigexpandtimes

Default value: true

trigexpandtimes controls the "product" rule for trigexpand. Thus, when the trigexpand command is used or the trigexpand switch set to true, expansion of products (e.g. sin(2*x)) will take place only if trigexpandtimes is true.

Option variable: triginverses

Default value: true

triginverses controls the simplification of the composition of trigonometric and hyperbolic functions with their inverse functions.

If all, both e.g. atan(tan(x)) and tan(atan(x)) simplify to x.

If true, the arcfun(fun(x)) simplification is turned off.

If false, both the arcfun(fun(x)) and fun(arcfun(x)) simplifications are turned off.

Function: trigreduce
    trigreduce (expr, x)
    trigreduce (expr)

Combines products and powers of trigonometric and hyperbolic sin’s and cos’s of x into those of multiples of x. It also tries to eliminate these functions when they occur in denominators. If x is omitted then all variables in expr are used.

See also poissimp.

(%i1) trigreduce(-sin(x)^2+3*cos(x)^2+x);
               cos(2 x)      cos(2 x)   1        1
(%o1)          -------- + 3 (-------- + -) + x - -
                  2             2       2        2
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.

Function: trigsimp (expr)

Employs the identities \(sin(x)^2 + cos(x)^2 = 1\) and \(cosh(x)^2 - sinh(x)^2 = 1\) to simplify expressions containing tan, sec, etc., to sin, cos, sinh, cosh.

trigreduce, ratsimp, and radcan may be able to further simplify the result.

demo ("trgsmp.dem") displays some examples of trigsimp.

Function: trigrat (expr)

Gives a canonical simplified quasilinear form of a trigonometrical expression; expr is a rational fraction of several sin, cos or tan, the arguments of them are linear forms in some variables (or kernels) and %pi/n (n integer) with integer coefficients. The result is a simplified fraction with numerator and denominator linear in sin and cos. Thus trigrat linearize always when it is possible.

(%i1) trigrat(sin(3*a)/sin(a+%pi/3));
(%o1)            sqrt(3) sin(2 a) + cos(2 a) - 1

The following example is taken from Davenport, Siret, and Tournier, Calcul Formel, Masson (or in English, Addison-Wesley), section 1.5.5, Morley theorem.

(%i1) c : %pi/3 - a - b$
(%i2) bc : sin(a)*sin(3*c)/sin(a+b);
                                           %pi
                 sin(a) sin(3 ((- b) - a + ---))
                                            3
(%o2)            -------------------------------
                           sin(b + a)
(%i3) ba : bc, c=a, a=c;
                                         %pi
                    sin(3 a) sin(b + a - ---)
                                          3
(%o3)               -------------------------
                                  %pi
                          sin(a - ---)
                                   3
(%i4) ac2 : ba^2 + bc^2 - 2*bc*ba*cos(b);
         2         2         %pi
      sin (3 a) sin (b + a - ---)
                              3
(%o4) ---------------------------
                2     %pi
             sin (a - ---)
                       3
                                         %pi
 - (2 sin(a) sin(3 a) sin(3 ((- b) - a + ---)) cos(b)
                                          3
             %pi            %pi
 sin(b + a - ---))/(sin(a - ---) sin(b + a))
              3              3
      2       2                %pi
   sin (a) sin (3 ((- b) - a + ---))
                                3
 + ---------------------------------
                 2
              sin (b + a)
(%i5) trigrat (ac2);
(%o5) - (sqrt(3) sin(4 b + 4 a) - cos(4 b + 4 a)
 - 2 sqrt(3) sin(4 b + 2 a) + 2 cos(4 b + 2 a)
 - 2 sqrt(3) sin(2 b + 4 a) + 2 cos(2 b + 4 a)
 + 4 sqrt(3) sin(2 b + 2 a) - 8 cos(2 b + 2 a) - 4 cos(2 b - 2 a)
 + sqrt(3) sin(4 b) - cos(4 b) - 2 sqrt(3) sin(2 b) + 10 cos(2 b)
 + sqrt(3) sin(4 a) - cos(4 a) - 2 sqrt(3) sin(2 a) + 10 cos(2 a)
 - 9)/4

Previous: , Up: Trigonometric Functions   [Contents][Index]

JavaScript license information