Trigonometric

Summary

Trigonometric
Functions
TrigonometricSubs	Substitute trigonometric variables with numerical values
TrigonometricAlgebraic	Convert trigonometric and radical expressions to algebraic ones
TrigonometricAlgebraicCosSin	Convert trigonometric and radical expressions to algebraic ones
TrigonometricVariables	Updates trigonometric variables to algebraic ones.
TrigonometricConstants	Returns the algebraic variables of trigonometric constants.
AlgebraicVariables	Convert algebraic variables to trigonometric ones.
AlgebraicTrigonometric	Convert algebraic systems back to trigonometric ones.
TrigonometricTanHalf	Parametrize trigonometric and radical expressions with tangent expression
TrigonometricGcd	Extract common factors of trigonometric expressions

Functions

TrigonometricSubs

TrigonometricSubs := proc (
    spec ::seq(name=algebraic),
    sys ::{algebraic, algebraic<algebraic, list({algebraic,algebraic=algebraic, algebraic<algebraic})},
    precision ::integer := 4,
    {
    noradical ::truefalse := false
    }
)

Substitute trigonometric variables with numerical values

Parameters

spec	sequence of equalities var=value where var is a name and value a numerical value.
formula	equation, inequation, polynomial or list of those, with trigonometric and algebraic expressions.
precision (optional)	integer: the number of digits that must be kept when modifying the values in spec.

Returns

The input formula where the name in spec have been replaced by the corresponding values; the algebraic relations satissfied by cos and sin are still valid with the chosen numerical values.

Exemple begin(code)

TrigonometricSubs (a=3, cos(a)^2+sin(a)^2);

1 end(code)

TrigonometricAlgebraic

TrigonometricAlgebraic := proc( sys ::list(algebraic),
{
norelation ::truefalse := false,
cossin ::{set,list}(name) := indets(sys,name),
tanhalf ::{set,list}(name) := [],
const ::truefalse := false
}    )

Convert trigonometric and radical expressions to algebraic ones

Parameters

sys	list of polynomials: the polynomial to convert
norelation = b (optional)	b is a boolean: if false then trigonometric relations are added to the output system; default value: false
cossin = l (optional)	l is a list of name: the variables to change into cosine and sine of the angle; default value: list of all the variables
tanhalf = l (optional)	l is a list of name: the variables to change into tan of half of the angle; default value: []

Returns

A sequence of 2 elements :

a list of polynomials: the new system with algebraic variables and new relations added at the beginning of the system
a table that has for indices the variables appearing in cos, sin and tan, and for entries the sequences of the corresponding algebraic variables.

Example

> sys, vars := TrigonometricAlgebraic ( [cos(x)-cos(y),cos(y)-z^3] );
                                                   3
                sys, vars := [cosx - cosy, cosy - z ], algvars
> vars[y];
                                     cosy

TrigonometricAlgebraicCosSin

TrigonometricAlgebraicCosSin := proc ( sys ::list(algebraic),
{
norelation ::truefalse := false,
variables ::{set,list}(name) := NULL,
const ::truefalse := false
}    )

Convert trigonometric and radical expressions to algebraic ones

Parameters

sys	list of polynomials: the polynomial to convert
norelation = b (optional)	b is a boolean: if false then trigonometric relations are added to the output system; default value: false
variables = l (optional)	l is a list of name: the variables to change; default value: all the variables

Returns

A sequence of 2 elements :

a list of polynomials: the new system with algebraic variables and new relations added at the beginning of the system
a table that has for indices the variables appearing in cos, sin and tan, and for entries the sequences of the corresponding algebraic variables.

Example

> sys, vars := TrigonometricAlgebraic ( [cos(x)-cos(y),cos(y)-z^3] );
                                                   3
                sys, vars := [cosx - cosy, cosy - z ], algvars
> vars[y];
                                     cosy

TrigonometricVariables

TrigonometricVariables := proc ( vars ::list(name),
algvars ::table )

Updates trigonometric variables to algebraic ones.

Parameters

vars	list of name: the variables to convert.
algvars	table returned by the TrigonometricAlgebraic or TrigonometricTanHalf function.

Returns

A list of names: the variables updated with new algebraic variables corresponding to the trigonometric functions.

TrigonometricConstants

TrigonometricConstants := proc ( algvars ::table )

Returns the algebraic variables of trigonometric constants.

Parameters

algvars

table returned by the TrigonometricAlgebraic or TrigonometricTanHalf function.

Returns

A list of names: the algebraic variables corresponding to the trigonometric constants.

AlgebraicVariables

AlgebraicVariables := proc ( vars ::list(name),
algvars ::table )

Convert algebraic variables to trigonometric ones.

Parameters

vars	list of name: the variables to convert.
algvars	table returned by the TrigonometricAlgebraic or TrigonometricTanHalf function.

Returns

A list of names: the original trigonometric variables corresponding to the algebraic variables.

AlgebraicTrigonometric

AlgebraicTrigonometric := proc ( algsys ::list(anything),
algvars ::table,
{
nosimplify ::truefalse := false
}    )

Convert algebraic systems back to trigonometric ones.

Parameters

algsys	list of polynomials: the polynomial to convert
algvars	table returned by TrigonometricAlgebraic or TrigonometricTanHalf.
nosimplify=b (optional)	b is a boolean: if true, the trigonometric expression are not simplified; default value: false.

Returns

A list of algebraic expressions: the system back with trigonometric functions.

Example

> sys,vars:=TrigonometricAlgebraic ( [cos(x)-cos(y),cos(y)-z^3] );
                                                   3
                sys, vars := [cosx - cosy, cosy - z ], algvars

> AlgebraicTrigonometric (sys,vars);
                                                    3
                        [cos(x) - cos(y), cos(y) - z ]

TrigonometricTanHalf

TrigonometricTanHalf := proc( sys ::list(algebraic),
{
generic ::truefalse := false,
variables ::{list,set}(name) := NULL,
const ::truefalse := false
}
   )

Parametrize trigonometric and radical expressions with tangent expression

Parameters

sys	list of polynomials: the polynomial to convert
generic = b (optional)	b is a boolean: if true, the angles are supposed different from Pi; default value: false.
variables = l (optional)	l is a list of name: the variables to change; default value: all the variables

Returns

a sequence of 2 elements:

a list of list of polynomial fractions: the union of several systems such that the union of there zeroes are the zeroes of the input system; if generic is true, then only a list of fractions is returned.
a table that has for indices the variables appearing in cos, sin and tan, and for entries the sequences of the corresponding algebraic variable.
new relations for the radical computations are added at the beginning of the system.

TrigonometricGcd

TrigonometricGcd := proc ( sys ::list(algebraic),
{
squarefree ::truefalse := true,
ordering ::list(name) := NULL
}
   )

Extract common factors of trigonometric expressions

Parameters

sys	list of trigonometric formula

Returns

a list of formula: the common factors of the formula in sys.