Previous: Introduction to Trigonometric, Up: Trigonometric Functions [Contents][Index]
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) ]
‘Category: Trigonometric functions’ ‘Category: Simplification flags and variables’
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)
‘Category: Trigonometric functions’ ‘Category: Hyperbolic functions’ ‘Category: Simplification flags and variables’
– Arc Cosine.
‘Category: Trigonometric functions’
– Hyperbolic Arc Cosine.
‘Category: Hyperbolic functions’
– Arc Cotangent.
‘Category: Trigonometric functions’
– Hyperbolic Arc Cotangent.
‘Category: Hyperbolic functions’
– Arc Cosecant.
‘Category: Trigonometric functions’
– Hyperbolic Arc Cosecant.
‘Category: Hyperbolic functions’
– Arc Secant.
‘Category: Trigonometric functions’
– Hyperbolic Arc Secant.
‘Category: Hyperbolic functions’
– Arc Sine.
‘Category: Trigonometric functions’
– Hyperbolic Arc Sine.
‘Category: Hyperbolic functions’
– yields the value of atan(y/x)
in the interval -%pi
to %pi
.
See also atan
.
‘Category: Trigonometric functions’
– Hyperbolic Arc Tangent.
‘Category: Hyperbolic functions’
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.
‘Category: Trigonometric functions’ ‘Category: Package atrig1’
– Cosine.
‘Category: Trigonometric functions’
– Hyperbolic Cosine.
‘Category: Hyperbolic functions’
– Cotangent.
‘Category: Trigonometric functions’
– Hyperbolic Cotangent.
‘Category: Hyperbolic functions’
– Cosecant.
‘Category: Trigonometric functions’
– Hyperbolic Cosecant.
‘Category: Hyperbolic functions’
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)
‘Category: Trigonometric functions’ ‘Category: Simplification flags and variables’
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
.
‘Category: Trigonometric functions’ ‘Category: Package ntrig’
– Secant.
‘Category: Trigonometric functions’
– Hyperbolic Secant.
‘Category: Hyperbolic functions’
– Sine.
‘Category: Trigonometric functions’
– Hyperbolic Sine.
‘Category: Hyperbolic functions’
– Tangent.
‘Category: Trigonometric functions’
– Hyperbolic Tangent.
‘Category: Hyperbolic functions’
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)
‘Category: Trigonometric functions’ ‘Category: Simplification functions’
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
.
‘Category: Trigonometric functions’ ‘Category: Simplification flags and variables’
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
.
‘Category: Trigonometric functions’ ‘Category: Simplification flags and variables’
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.
‘Category: Trigonometric functions’ ‘Category: Simplification flags and variables’
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
‘Category: Trigonometric functions’ ‘Category: Simplification functions’
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
.
‘Category: Trigonometric functions’ ‘Category: Simplification flags and variables’
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
.
‘Category: Trigonometric functions’ ‘Category: Simplification functions’
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
‘Category: Trigonometric functions’ ‘Category: Simplification functions’
Previous: Introduction to Trigonometric, Up: Trigonometric Functions [Contents][Index]