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) ]
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)
– Arc Cosine.
– Hyperbolic Arc Cosine.
– Arc Cotangent.
– Hyperbolic Arc Cotangent.
– Arc Cosecant.
– Hyperbolic Arc Cosecant.
– Arc Secant.
– Hyperbolic Arc Secant.
– Arc Sine.
– Hyperbolic Arc Sine.
– Arc Tangent.
See also atan2.
– yields the value of atan(y/x) in the interval -%pi
to %pi.
See also atan.
– Hyperbolic Arc Tangent.
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.
– Cosine.
– Hyperbolic Cosine.
– Cotangent.
– Hyperbolic Cotangent.
– Cosecant.
– Hyperbolic Cosecant.
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)
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.
– Secant.
– Hyperbolic Secant.
– Sine.
– Hyperbolic Sine.
– Tangent.
– Hyperbolic Tangent.
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:
trigexpandIf true causes expansion of all
expressions containing sin’s and cos’s occurring subsequently.
halfanglesIf true causes half-angles to be simplified
away.
trigexpandplusControls the "sum" rule for trigexpand,
expansion of sums (e.g. sin(x + y)) will take place only if
trigexpandplus is true.
trigexpandtimesControls 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)
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.
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.
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.
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
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.
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.
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: Introduction to Trigonometric, Up: Trigonometric Functions [Contents][Index]