MathAction SandBoxSinCosRules

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Sin and Cos Rules

fricas

x:=v1*sin(p1)+v2*sin(p2)

$\label{eq1}{v 2 \ {\sin \left({p 2}\right)}}+{v 1 \ {\sin \left({p 1}\right)}}$

(1)

Type: Expression(Integer)

fricas

y:= a0 + a1*x + a2*x^2 + a3*x^3

$\label{eq2}\begin{array}{@{}l} \displaystyle {a 3 \ {{v 2}^{3}}\ {{\sin \left({p 2}\right)}^{3}}}+{{\left({3 \ a 3 \ v 1 \ {{v 2}^{2}}\ {\sin \left({p 1}\right)}}+{a 2 \ {{v 2}^{2}}}\right)}\ {{\sin \left({p 2}\right)}^{2}}}+ \ \ \displaystyle {{\left({3 \ a 3 \ {{v 1}^{2}}\ v 2 \ {{\sin \left({p 1}\right)}^{2}}}+{2 \ a 2 \ v 1 \ v 2 \ {\sin \left({p 1}\right)}}+{a 1 \ v 2}\right)}\ {\sin \left({p 2}\right)}}+ \ \ \displaystyle {a 3 \ {{v 1}^{3}}\ {{\sin \left({p 1}\right)}^{3}}}+{a 2 \ {{v 1}^{2}}\ {{\sin \left({p 1}\right)}^{2}}}+{a 1 \ v 1 \ {\sin \left({p 1}\right)}}+ a 0$

(2)

Type: Expression(Integer)

fricas

sinCosProducts := rule
  sin(x)*sin(y) == (cos(x-y) - cos(x+y))/2
  cos(x)*cos(y) == (cos(x-y) + cos(x+y))/2
  sin(x)*cos(y) == (sin(x-y) + sin(x+y))/2
  sin(x)^2 == (1 - cos(2*x))/2
  sin(x)^3 == sin(x)*(1 - cos(2*x))/2

$\label{eq3}\begin{array}{@{}l} \displaystyle \left\{{= = \left({{\%B \ {\sin \left({x}\right)}\ {\sin \left({y}\right)}}, \:{{-{\%B \ {\cos \left({y + x}\right)}}+{\%B \ {\cos \left({y - x}\right)}}}\over 2}}\right)}, \right. \ \ \displaystyle \left.\:{= = \left({{\%C \ {\cos \left({x}\right)}\ {\cos \left({y}\right)}}, \:{{{\%C \ {\cos \left({y + x}\right)}}+{\%C \ {\cos \left({y - x}\right)}}}\over 2}}\right)}, \: \right. \ \ \displaystyle \left.{= = \left({{\%D \ {\cos \left({y}\right)}\ {\sin \left({x}\right)}}, \:{{{\%D \ {\sin \left({y + x}\right)}}-{\%D \ {\sin \left({y - x}\right)}}}\over 2}}\right)}, \: \right. \ \ \displaystyle \left.{= = \left({{{\sin \left({x}\right)}^{2}}, \:{{-{\cos \left({2 \ x}\right)}+ 1}\over 2}}\right)}, \: \right. \ \ \displaystyle \left.{= = \left({{{\sin \left({x}\right)}^{3}}, \:{{{\left(-{\cos \left({2 \ x}\right)}+ 1 \right)}\ {\sin \left({x}\right)}}\over 2}}\right)}\right\}$

(3)

Type: Ruleset(Integer,Integer,Expression(Integer))

fricas

sinCosProducts(y)

$\label{eq4}{\left( \begin{array}{@{}l} \displaystyle -{a 3 \ {{v 2}^{3}}\ {\sin \left({3 \ p 2}\right)}}-{3 \ a 3 \ v 1 \ {{v 2}^{2}}\ {\sin \left({{2 \ p 2}+ p 1}\right)}}+ \ \ \displaystyle {3 \ a 3 \ v 1 \ {{v 2}^{2}}\ {\sin \left({{2 \ p 2}- p 1}\right)}}- \ \ \displaystyle {3 \ a 3 \ {{v 1}^{2}}\ v 2 \ {\sin \left({p 2 +{2 \ p 1}}\right)}}+ \ \ \displaystyle {{\left({3 \ a 3 \ {{v 2}^{3}}}+{{\left({6 \ a 3 \ {{v 1}^{2}}}+{4 \ a 1}\right)}\ v 2}\right)}\ {\sin \left({p 2}\right)}}- \ \ \displaystyle {3 \ a 3 \ {{v 1}^{2}}\ v 2 \ {\sin \left({p 2 -{2 \ p 1}}\right)}}-{a 3 \ {{v 1}^{3}}\ {\sin \left({3 \ p 1}\right)}}+ \ \ \displaystyle {{\left({6 \ a 3 \ v 1 \ {{v 2}^{2}}}+{3 \ a 3 \ {{v 1}^{3}}}+{4 \ a 1 \ v 1}\right)}\ {\sin \left({p 1}\right)}}- \ \ \displaystyle {2 \ a 2 \ {{v 2}^{2}}\ {\cos \left({2 \ p 2}\right)}}-{4 \ a 2 \ v 1 \ v 2 \ {\cos \left({p 2 + p 1}\right)}}+ \ \ \displaystyle {4 \ a 2 \ v 1 \ v 2 \ {\cos \left({p 2 - p 1}\right)}}-{2 \ a 2 \ {{v 1}^{2}}\ {\cos \left({2 \ p 1}\right)}}+ \ \ \displaystyle {2 \ a 2 \ {{v 2}^{2}}}+{2 \ a 2 \ {{v 1}^{2}}}+{4 \ a 0}$

(4)

Type: Expression(Integer)

Integration

fricas

)clear completely

   All user variables and function definitions have been cleared.
   All )browse facility databases have been cleared.
   Internally cached functions and constructors have been cleared.
   )clear completely is finished.
integrate(%e ^(-x*x), x=%minusInfinity..%plusInfinity)

$\label{eq5}\sqrt{\pi}$

(5)

Type: Union(f1: OrderedCompletion?(Expression(Integer)),...)

fricas

integrate(integrate(x+y, x),y)

$\label{eq6}{{1 \over 2}\ x \ {{y}^{2}}}+{{1 \over 2}\ {{x}^{2}}\ y}$

(6)

Type: Polynomial(Fraction(Integer))

fricas

integrate(exp(-b*h*sin(o))*sin(o),o=0..2*%pi)

$\label{eq7}\verb#"failed"#$

(7)

Type: Union(fail: failed,...)

Gosper's algorithm is implemented in Axiom

fricas

p:Polynomial Integer:=1+x+x^2

$\label{eq8}{{x}^{2}}+ x + 1$

(8)

Type: Polynomial(Integer)

fricas

sum(p,x=0..n)

$\label{eq9}{{{n}^{3}}+{3 \ {{n}^{2}}}+{5 \ n}+ 3}\over 3$

(9)

Type: Fraction(Polynomial(Integer))

fricas

e:=binomial(2*k,k)/4^k

$\label{eq10}{\hbox{\axiomType{BINOMIAL}\ } \left({{2 \ k}, \: k}\right)}\over{{4}^{k}}$

(10)

Type: Expression(Integer)

fricas

sum(e,k=0..n)

$\label{eq11}{{\left({2 \ n}+ 1 \right)}\ {\hbox{\axiomType{BINOMIAL}\ } \left({{2 \ n}, \: n}\right)}}\over{{4}^{n}}$

(11)

Type: Expression(Integer)

fricas

sum(%,n=0..n)

$\label{eq12}{{\left({4 \ {{n}^{2}}}+{8 \ n}+ 3 \right)}\ {\hbox{\axiomType{BINOMIAL}\ } \left({{2 \ n}, \: n}\right)}}\over{3 \ {{4}^{n}}}$

(12)

Type: Expression(Integer)

fricas

factor(4*n^2+8*n+3)

$\label{eq13}{\left({2 \ n}+ 1 \right)}\ {\left({2 \ n}+ 3 \right)}$

(13)

Type: Factored(Polynomial(Integer))

Examples from Petkovsek, Wilf, Zeilberger

exp(log(x)+log(y^-1));

reduce

$\displaylines{\qdd \frac{x}{ y} \cr}$

Symbols, kernels, variables, expressions ... difficult to understand

fricas

e0:Expression Integer:=1+2*x^2+x

$\label{eq14}{2 \ {{x}^{2}}}+ x + 1$

(14)

Type: Expression(Integer)

fricas

kernels(e0)

$\label{eq15}\left[ x \right]$

(15)

Type: List(Kernel(Expression(Integer)))

fricas

p0:=e0::(Polynomial Integer)

$\label{eq16}{2 \ {{x}^{2}}}+ x + 1$

(16)

Type: Polynomial(Integer)

fricas

variables(p0)

$\label{eq17}\left[ x \right]$

(17)

Type: List(Symbol)

fricas

e1:Expression Integer:=x*sin(t)/cos(t)+1

$\label{eq18}{{x \ {\sin \left({t}\right)}}+{\cos \left({t}\right)}}\over{\cos \left({t}\right)}$

(18)

Type: Expression(Integer)

fricas

kernels(e1)

$\label{eq19}\left[{\sin \left({t}\right)}, \:{\cos \left({t}\right)}, \: x \right]$

(19)

Type: List(Kernel(Expression(Integer)))

fricas

solve(e1=0,x)

$\label{eq20}\left[{x = -{{\cos \left({t}\right)}\over{\sin \left({t}\right)}}}\right]$

(20)

Type: List(Equation(Expression(Integer)))

fricas

e2:UP(x,Expression Integer):=sin(t)/cos(t)*x

$\label{eq21}{{\sin \left({t}\right)}\over{\cos \left({t}\right)}}\ x$

(21)

Type: UnivariatePolynomial(x,Expression(Integer))

Expressions and substitution

fricas

f:=operator 'f; e:=1+a*x^2+f(y)*x^3; eq:=[f(y)=r]; peq:=subst(e,eq)::(Polynomial Integer)=0;sol:=radicalSolve(peq,x);

Type: List(Equation(Expression(Integer)))

fricas

--[(lhs(s)=subst(rhs(s),r=f(y))) for s in sol]
--As a list there is no output shown
--x1=subst (rhs(sol.1),r=f(y))
--x2=subst (rhs(sol.2),r=f(y))
x3=subst (rhs(sol.3),r=f(y))

$\label{eq22}\begin{array}{@{}l} \displaystyle x 3 ={{\left( \begin{array}{@{}l} \displaystyle { \begin{array}{@{}l} \displaystyle 9 \ {{f \left({y}\right)}^{2}}\ \cdot \ \ \displaystyle {{\root{3}\of{{{{54}\ {{f \left({y}\right)}^{3}}\ {\sqrt{{{{2 7}\ {{f \left({y}\right)}^{2}}}+{4 \ {{a}^{3}}}}\over{{108}\ {{f \left({y}\right)}^{4}}}}}}-{{27}\ {{f \left({y}\right)}^{2}}}-{2 \ {{a}^{3}}}}\over{{54}\ {{f \left({y}\right)}^{3}}}}}^{2}}$