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| 15.1 Introduction to Trigonometric | ||
| 15.2 Functions and Variables 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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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;
(%o1) 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;
(%o4) 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);
(%o1) done
(%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; (%o1) false (%i2) [sin (%i * x), cos (%i * x), tan (%i * x)]; (%o2) [sin(%i x), cos(%i x), tan(%i x)] (%i3) %iargs : true; (%o3) 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); (%o1) done (%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.
- 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:
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: 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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