Elementary Functions
%iargs — Variable
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:
maxima
(%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.
maxima
(%i1) declare (x, imaginary)$
(%i2) [featurep (x, imaginary), featurep (x, real)];
(%o2) [true, false]
(%i3) sin (%i * x);
(%o3) %i sinh(x)
%piargs — Variable
Default value: true
When %piargs is true,
trigonometric functions are simplified to algebraic constants
when the argument is an integer multiple
of
$\pi,$
$\pi/2,$
$\pi/4,$
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:
maxima
(%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.
maxima
(%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) ]
acos (x) — Function
– Arc Cosine.
For variables that control simplification _0025piargs,
_0025iargs, halfangles, triginverses, and
trigsign.
See also: %piargs, %iargs, halfangles, triginverses, trigsign.
acosh (x) — Function
– Hyperbolic Arc Cosine.
For variables that control simplification _0025piargs,
_0025iargs, halfangles, triginverses, and
trigsign.
See also: %piargs, %iargs, halfangles, triginverses, trigsign.
acot (x) — Function
– Arc Cotangent.
For variables that control simplification _0025piargs,
_0025iargs, halfangles, triginverses, and
trigsign.
See also: %piargs, %iargs, halfangles, triginverses, trigsign.
acoth (x) — Function
– Hyperbolic Arc Cotangent.
For variables that control simplification _0025piargs,
_0025iargs, halfangles, triginverses, and
trigsign.
See also: %piargs, %iargs, halfangles, triginverses, trigsign.
acsc (x) — Function
– Arc Cosecant.
For variables that control simplification _0025piargs,
_0025iargs, halfangles, triginverses, and
trigsign.
See also: %piargs, %iargs, halfangles, triginverses, trigsign.
acsch (x) — Function
– Hyperbolic Arc Cosecant.
For variables that control simplification _0025piargs,
_0025iargs, halfangles, triginverses, and
trigsign.
See also: %piargs, %iargs, halfangles, triginverses, trigsign.
asec (x) — Function
– Arc Secant.
For variables that control simplification _0025piargs,
_0025iargs, halfangles, triginverses, and
trigsign.
See also: %piargs, %iargs, halfangles, triginverses, trigsign.
asech (x) — Function
– Hyperbolic Arc Secant.
For variables that control simplification _0025piargs,
_0025iargs, halfangles, triginverses, and
trigsign.
See also: %piargs, %iargs, halfangles, triginverses, trigsign.
asin (x) — Function
– Arc Sine.
For variables that control simplification _0025piargs,
_0025iargs, halfangles, triginverses, and
trigsign.
See also: %piargs, %iargs, halfangles, triginverses, trigsign.
asinh (x) — Function
– Hyperbolic Arc Sine.
For variables that control simplification _0025piargs,
_0025iargs, halfangles, triginverses, and
trigsign.
See also: %piargs, %iargs, halfangles, triginverses, trigsign.
atan (x) — Function
– Arc Tangent.
See also atan2.
For variables that control simplification _0025piargs,
_0025iargs, halfangles, triginverses, and
trigsign.
See also: atan2, %piargs, %iargs, halfangles, triginverses, trigsign.
atan2 (y, x) — Function
– yields the value of $\tan^{-1}(y/x)$ in the interval $-\pi$ to $\pi$ taking into consideration the quadrant of the point $(x,y).$
Along the branch cut with $y = 0$ and $x < 0$, atan2
is continuous with the second quadrant.
See also atan.
For variables that control simplification _0025piargs,
_0025iargs, halfangles, triginverses, and
trigsign.
See also: atan, %piargs, %iargs, halfangles, triginverses, trigsign.
atanh (x) — Function
– Hyperbolic Arc Tangent.
For variables that control simplification _0025piargs,
_0025iargs, halfangles, triginverses, and
trigsign.
See also: %piargs, %iargs, halfangles, triginverses, trigsign.
atrig1 — Variable
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.
cos (x) — Function
– Cosine.
For variables that control simplification _0025piargs,
_0025iargs, halfangles, triginverses, and
trigsign.
See also: %piargs, %iargs, halfangles, triginverses, trigsign.
cosh (x) — Function
– Hyperbolic Cosine.
For variables that control simplification _0025piargs,
_0025iargs, halfangles, triginverses, and
trigsign.
See also: %piargs, %iargs, halfangles, triginverses, trigsign.
cot (x) — Function
– Cotangent.
For variables that control simplification _0025piargs,
_0025iargs, halfangles, triginverses, and
trigsign.
See also: %piargs, %iargs, halfangles, triginverses, trigsign.
coth (x) — Function
– Hyperbolic Cotangent.
For variables that control simplification _0025piargs,
_0025iargs, halfangles, triginverses, and
trigsign.
See also: %piargs, %iargs, halfangles, triginverses, trigsign.
csc (x) — Function
– Cosecant.
For variables that control simplification _0025piargs,
_0025iargs, halfangles, triginverses, and
trigsign.
See also: %piargs, %iargs, halfangles, triginverses, trigsign.
csch (x) — Function
– Hyperbolic Cosecant.
For variables that control simplification _0025piargs,
_0025iargs, halfangles, triginverses, and
trigsign.
See also: %piargs, %iargs, halfangles, triginverses, trigsign.
halfangles — Variable
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 $0 \le x < 2\pi,$ $\sin{x\over 2}$ simplifies to a simple formula:
$${\sqrt{1-\cos x}\over\sqrt{2}}$$
$${\sqrt{1-\cos x}\over\sqrt{2}}$$
A complicated factor is needed to make this formula correct for all complex arguments $z = x+iy$:
$$(-1)^{\lfloor{x/(2\pi)}\rfloor} \left[1-\rm{unit_step}(-y) \left(1+(-1)^{\lfloor{x/(2\pi)}\rfloor - \lceil{x/(2\pi)}\rceil}\right)\right]$$
$$(-1)^{\lfloor{x/(2\pi)}\rfloor} \left[1-\rm{unit_step}(-y) \left(1+(-1)^{\lfloor{x/(2\pi)}\rfloor - \lceil{x/(2\pi)}\rceil}\right)\right] $$
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:
maxima
(%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)
ntrig — Variable
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.
sec (x) — Function
– Secant.
For variables that control simplification _0025piargs,
_0025iargs, halfangles, triginverses, and
trigsign.
See also: %piargs, %iargs, halfangles, triginverses, trigsign.
sech (x) — Function
– Hyperbolic Secant.
For variables that control simplification _0025piargs,
_0025iargs, halfangles, triginverses, and
trigsign.
See also: %piargs, %iargs, halfangles, triginverses, trigsign.
sin (x) — Function
– Sine.
For variables that control simplification _0025piargs,
_0025iargs, halfangles, triginverses, and
trigsign.
See also: %piargs, %iargs, halfangles, triginverses, trigsign.
sinh (x) — Function
– Hyperbolic Sine.
For variables that control simplification _0025piargs,
_0025iargs, halfangles, triginverses, and
trigsign.
See also: %piargs, %iargs, halfangles, triginverses, trigsign.
tan (x) — Function
– Tangent.
For variables that control simplification _0025piargs,
_0025iargs, halfangles, triginverses, and
trigsign.
See also: %piargs, %iargs, halfangles, triginverses, trigsign.
tanh (x) — Function
– Hyperbolic Tangent.
For variables that control simplification _0025piargs,
_0025iargs, halfangles, triginverses, and
trigsign.
See also: %piargs, %iargs, halfangles, triginverses, trigsign.
trigexpand (expr) — Function
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:
maxima
(%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)
See also: trigexpand, halfangles, trigexpandplus, trigexpandtimes.
trigexpandplus — Variable
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.
See also: trigexpand.
trigexpandtimes — Variable
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.
See also: trigexpand.
triginverses — Variable
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.
trigrat (expr) — Function
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.
maxima
(%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.
maxima
(%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
trigreduce (expr, x) — Function
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.
maxima
(%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
See also: poissimp.
trigsign — Variable
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.
trigsimp (expr) — Function
Employs the identities $\sin\left(x\right)^2 + \cos\left(x\right)^2 = 1$ and $\cosh\left(x\right)^2 - \sinh\left(x\right)^2 = 1$ 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.
See also: trigreduce, ratsimp, radcan.