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15.1 Introduction to Trigonometric | ||
15.2 Definitions for Trigonometric |
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Maxima has many trigonometric functions defined. Not all trigonometric
identities are programmed, but it is possible for the user to add many
of them using the pattern matching capabilities of the system. The
trigonometric functions defined in Maxima are: acos
,
acosh
, acot
, acoth
, acsc
,
acsch
, asec
, asech
, asin
,
asinh
, atan
, atanh
, cos
,
cosh
, cot
, coth
, csc
, csch
,
sec
, sech
, sin
, sinh
, tan
,
and tanh
. There are a number of commands especially for
handling trigonometric functions, see trigexpand
,
trigreduce
, and the switch trigsign
. Two share
packages extend the simplification rules built into Maxima,
ntrig
and atrig1
. Do describe(command)
for details.
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- 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.
- yields the value of atan(y/x)
in the interval -%pi
to
%pi
.
- 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
,
half-angles are simplified away.
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:
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) |
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: all
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 |
The trigonometric simplification routines will use declared information in some simple cases. Declarations about variables are used as follows, e.g.
(%i1) declare(j, integer, e, even, o, odd)$ (%i2) sin(x + (e + 1/2)*%pi); (%o2) cos(x) (%i3) sin(x + (o + 1/2)*%pi); (%o3) - cos(x) |
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 simplifyed 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; %pi (%o1) - b - a + --- 3 (%i2) bc: sin(a)*sin(3*c)/sin(a+b); sin(a) sin(3 b + 3 a) (%o2) --------------------- sin(b + a) (%i3) ba: bc, c=a, a=c$ (%i4) ac2: ba^2 + bc^2 - 2*bc*ba*cos(b); 2 2 sin (a) sin (3 b + 3 a) (%o4) ----------------------- 2 sin (b + a) %pi 2 sin(a) sin(3 a) cos(b) sin(b + a - ---) sin(3 b + 3 a) 3 - -------------------------------------------------------- %pi sin(a - ---) sin(b + a) 3 2 2 %pi sin (3 a) sin (b + a - ---) 3 + --------------------------- 2 %pi sin (a - ---) 3 (%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 |
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