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ExampleTrigonometricConstants

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Example of computations with trigonometric constants

Consider the problem:

Let $t := (5/9)\pi$.  Simplify $tan(t) + 4sin(t)$.

We can solve it by noting that involved numbers can be expressed in terms of roots of $1$,  Namely,
first we express trigonometric functions in terms of complex exponentials.  Since $(5/9)\pi$ is
a rational multiple of $\pi$ we get root of $1$, in this case root of degree $18$.  But we also
need $i$, which leads to root of degree $36$. Corresponding field is extention by cyclotomic
polynomial, so we build it:
\begin{axiom}
cyclotomic(36)$CyclotomicUtilities
)set output algebra on
)set output tex off
cF := SAE(FRAC(INT), SUP(FRAC(INT)), %)
)set output tex on
)set output algebra off
\end{axiom}
Now, we can express all ingredients in terms of generator:
\begin{axiom}
et := (generator()$cF)^10
ct := (et + 1/et)/2
st := (et - 1/et)/(2*(generator()$cF)^9)
\end{axiom}
and our expression is:
\begin{axiom}
ex := (4*ct + 1)*st/ct
\end{axiom}
Now we can find minimal polynomial and find its roots:
\begin{axiom}
minimalPolynomial(%)
\end{axiom}
and we see that the result is $\sqrt(3)$.

We could also observe that the expression above is a member of smaller cyclotomic field and from           
that infer expression $2\cos(\pi/6)$.

Consder now expression
$$
i(5\sin((5\pi)/7)-5\sin((4\pi)/7)+5\sin((3\pi)/7) -5\sin((2\pi)/7)) -5\cos((5\pi)/7)+5\cos((4\pi)/7)
$$
$$
-5\cos((3\pi)/7)+5\cos((2\pi)/7) -10\cos(\pi/7)+6.
$$

Now we need root of $1$ of degree $28$, so we build the field:
\begin{axiom}
cyclotomic(28)$CyclotomicUtilities
)set output algebra on
)set output tex off
cF2 := SAE(FRAC(INT), SUP(FRAC(INT)), %)
)set output tex on
)set output algebra off
\end{axiom}

Root of degree $4$ that is generator to power $7$ gives us imaginary unit $i$,  Suare of generator gives
us primitive root of degree $7$, that is $\exp(\pi/7)$:
\begin{axiom}
im2 := generator()$cF2^7
e1 := generator()$cF2^2
\end{axiom}
We express trigonometric functions in terms of this
\begin{axiom}
c1 := (e1 + 1/e1)/2
e2 := e1^2
c2 := (e2 + 1/e2)/2
s2 := (e2 - 1/e2)/(2*im2)
e3 := e1^3
c3 := (e3 + 1/e3)/2
s3 := (e3 - 1/e3)/(2*im2)
e4 := e1^4
c4 := (e4 + 1/e4)/2
s4 := (e4 - 1/e4)/(2*im2)
e5 := e1^5
c5 := (e5 + 1/e5)/2
s5 := (e5 - 1/e5)/(2*im2)
\end{axiom}
Using this we can compute the expression:
\begin{axiom}
im2*(5*s5 - 5*s4 + 5*s3 - 5*s2) - 5*c5 + 5*c4 - 5*c3 + 5*c2 - 10*c1 + 6
\end{axiom}

Example of computations with trigonometric constants

Consider the problem:

Let $t := (5/9)\pi$ . Simplify .

We can solve it by noting that involved numbers can be expressed in terms of roots of , Namely, first we express trigonometric functions in terms of complex exponentials. Since $(5/9)\pi$ is a rational multiple of $\pi$ we get root of , in this case root of degree . But we also need , which leads to root of degree . Corresponding field is extention by cyclotomic polynomial, so we build it:

fricas

(1) -> cyclotomic(36)$CyclotomicUtilities

$\label{eq1}{{?}^{12}}-{{?}^{6}}+ 1$

(1)

Type: SparseUnivariatePolynomial?(Integer)

fricas

)set output algebra on

fricas

)set output tex off

cF := SAE(FRAC(INT), SUP(FRAC(INT)), %)

   (2)
   SimpleAlgebraicExtension
      Fraction(Integer)
  ,
      SparseUnivariatePolynomial(Fraction(Integer))
  ,
       12    6
      ?   - ?  + 1

Type: Type

fricas

)set output tex on

fricas

)set output algebra off

Now, we can express all ingredients in terms of generator:

fricas

et := (generator()$cF)^10

$\label{eq2}{?}^{10}$

(2)

Type: SimpleAlgebraicExtension?(Fraction(Integer),SparseUnivariatePolynomial?(Fraction(Integer)),?^12+(-(?^6))+1)

fricas

ct := (et + 1/et)/2

$\label{eq3}{{\frac{1}{2}}\ {{?}^{10}}}-{{\frac{1}{2}}\ {{?}^{8}}}$

(3)

Type: SimpleAlgebraicExtension?(Fraction(Integer),SparseUnivariatePolynomial?(Fraction(Integer)),?^12+(-(?^6))+1)

fricas

st := (et - 1/et)/(2*(generator()$cF)^9)

$\label{eq4}-{{\frac{1}{2}}\ {{?}^{11}}}+{{\frac{1}{2}}\ {{?}^{5}}}+{{\frac{1}{2}}\ ?}$

(4)

Type: SimpleAlgebraicExtension?(Fraction(Integer),SparseUnivariatePolynomial?(Fraction(Integer)),?^12+(-(?^6))+1)

and our expression is:

fricas

ex := (4*ct + 1)*st/ct

$\label{eq5}{{?}^{9}}-{2 \ {{?}^{3}}}$

(5)

Type: SimpleAlgebraicExtension?(Fraction(Integer),SparseUnivariatePolynomial?(Fraction(Integer)),?^12+(-(?^6))+1)

Now we can find minimal polynomial and find its roots:

fricas

minimalPolynomial(%)

$\label{eq6}{{?}^{2}}- 3$

(6)

Type: SparseUnivariatePolynomial?(Fraction(Integer))

and we see that the result is $\sqrt(3)$ .

We could also observe that the expression above is a member of smaller cyclotomic field and from that infer expression $2\cos(\pi/6)$ .

Consder now expression

$i(5\sin((5\pi)/7)-5\sin((4\pi)/7)+5\sin((3\pi)/7) -5\sin((2\pi)/7)) -5\cos((5\pi)/7)+5\cos((4\pi)/7)$

$-5\cos((3\pi)/7)+5\cos((2\pi)/7) -10\cos(\pi/7)+6.$

Now we need root of of degree , so we build the field:

fricas

cyclotomic(28)$CyclotomicUtilities

$\label{eq7}{{?}^{12}}-{{?}^{10}}+{{?}^{8}}-{{?}^{6}}+{{?}^{4}}-{{?}^{2}}+ 1$

(7)

Type: SparseUnivariatePolynomial?(Integer)

fricas

)set output algebra on

fricas

)set output tex off

cF2 := SAE(FRAC(INT), SUP(FRAC(INT)), %)

   (9)
   SimpleAlgebraicExtension
      Fraction(Integer)
  ,
      SparseUnivariatePolynomial(Fraction(Integer))
  ,
       12    10    8    6    4    2
      ?   - ?   + ?  - ?  + ?  - ?  + 1

Type: Type

fricas

)set output tex on

fricas

)set output algebra off

Root of degree that is generator to power gives us imaginary unit , Suare of generator gives us primitive root of degree , that is $\exp(\pi/7)$ :

fricas

im2 := generator()$cF2^7

$\label{eq8}{?}^{7}$

(8)

Type: SimpleAlgebraicExtension?(Fraction(Integer),SparseUnivariatePolynomial?(Fraction(Integer)),?^12+(-(?^10))+?^8+(-(?^6))+?^4+(-(?^2))+1)

fricas

e1 := generator()$cF2^2

$\label{eq9}{?}^{2}$

(9)

Type: SimpleAlgebraicExtension?(Fraction(Integer),SparseUnivariatePolynomial?(Fraction(Integer)),?^12+(-(?^10))+?^8+(-(?^6))+?^4+(-(?^2))+1)

We express trigonometric functions in terms of this

fricas

c1 := (e1 + 1/e1)/2

$\label{eq10}-{{\frac{1}{2}}\ {{?}^{10}}}+{{\frac{1}{2}}\ {{?}^{8}}}-{{\frac{1}{2}}\ {{?}^{6}}}+{{\frac{1}{2}}\ {{?}^{4}}}+{\frac{1}{2}}$

(10)

Type: SimpleAlgebraicExtension?(Fraction(Integer),SparseUnivariatePolynomial?(Fraction(Integer)),?^12+(-(?^10))+?^8+(-(?^6))+?^4+(-(?^2))+1)

fricas

e2 := e1^2

$\label{eq11}{?}^{4}$

(11)

Type: SimpleAlgebraicExtension?(Fraction(Integer),SparseUnivariatePolynomial?(Fraction(Integer)),?^12+(-(?^10))+?^8+(-(?^6))+?^4+(-(?^2))+1)

fricas

c2 := (e2 + 1/e2)/2

$\label{eq12}-{{\frac{1}{2}}\ {{?}^{10}}}+{{\frac{1}{2}}\ {{?}^{4}}}$

(12)

Type: SimpleAlgebraicExtension?(Fraction(Integer),SparseUnivariatePolynomial?(Fraction(Integer)),?^12+(-(?^10))+?^8+(-(?^6))+?^4+(-(?^2))+1)

fricas

s2 := (e2 - 1/e2)/(2*im2)

$\label{eq13}-{{\frac{1}{2}}\ {{?}^{11}}}+{{\frac{1}{2}}\ {{?}^{3}}}$

(13)

Type: SimpleAlgebraicExtension?(Fraction(Integer),SparseUnivariatePolynomial?(Fraction(Integer)),?^12+(-(?^10))+?^8+(-(?^6))+?^4+(-(?^2))+1)

fricas

e3 := e1^3

$\label{eq14}{?}^{6}$

(14)

Type: SimpleAlgebraicExtension?(Fraction(Integer),SparseUnivariatePolynomial?(Fraction(Integer)),?^12+(-(?^10))+?^8+(-(?^6))+?^4+(-(?^2))+1)

fricas

c3 := (e3 + 1/e3)/2

$\label{eq15}-{{\frac{1}{2}}\ {{?}^{8}}}+{{\frac{1}{2}}\ {{?}^{6}}}$

(15)

Type: SimpleAlgebraicExtension?(Fraction(Integer),SparseUnivariatePolynomial?(Fraction(Integer)),?^12+(-(?^10))+?^8+(-(?^6))+?^4+(-(?^2))+1)

fricas

s3 := (e3 - 1/e3)/(2*im2)

$\label{eq16}\begin{array}{@{}l} \displaystyle -{{\frac{1}{2}}\ {{?}^{11}}}+{{\frac{1}{2}}\ {{?}^{9}}}-{{\frac{1}{2}}\ {{?}^{7}}}+{{\frac{1}{2}}\ {{?}^{5}}}- \ \ \displaystyle {{\frac{1}{2}}\ {{?}^{3}}}+ ?$

(16)

Type: SimpleAlgebraicExtension?(Fraction(Integer),SparseUnivariatePolynomial?(Fraction(Integer)),?^12+(-(?^10))+?^8+(-(?^6))+?^4+(-(?^2))+1)

fricas

e4 := e1^4

$\label{eq17}{?}^{8}$

(17)

Type: SimpleAlgebraicExtension?(Fraction(Integer),SparseUnivariatePolynomial?(Fraction(Integer)),?^12+(-(?^10))+?^8+(-(?^6))+?^4+(-(?^2))+1)

fricas

c4 := (e4 + 1/e4)/2

$\label{eq18}{{\frac{1}{2}}\ {{?}^{8}}}-{{\frac{1}{2}}\ {{?}^{6}}}$

(18)

Type: SimpleAlgebraicExtension?(Fraction(Integer),SparseUnivariatePolynomial?(Fraction(Integer)),?^12+(-(?^10))+?^8+(-(?^6))+?^4+(-(?^2))+1)

fricas

s4 := (e4 - 1/e4)/(2*im2)

$\label{eq19}\begin{array}{@{}l} \displaystyle -{{\frac{1}{2}}\ {{?}^{11}}}+{{\frac{1}{2}}\ {{?}^{9}}}-{{\frac{1}{2}}\ {{?}^{7}}}+{{\frac{1}{2}}\ {{?}^{5}}}- \ \ \displaystyle {{\frac{1}{2}}\ {{?}^{3}}}+ ?$

(19)

Type: SimpleAlgebraicExtension?(Fraction(Integer),SparseUnivariatePolynomial?(Fraction(Integer)),?^12+(-(?^10))+?^8+(-(?^6))+?^4+(-(?^2))+1)

fricas

e5 := e1^5

$\label{eq20}{?}^{10}$

(20)

Type: SimpleAlgebraicExtension?(Fraction(Integer),SparseUnivariatePolynomial?(Fraction(Integer)),?^12+(-(?^10))+?^8+(-(?^6))+?^4+(-(?^2))+1)

fricas

c5 := (e5 + 1/e5)/2

$\label{eq21}{{\frac{1}{2}}\ {{?}^{10}}}-{{\frac{1}{2}}\ {{?}^{4}}}$

(21)

Type: SimpleAlgebraicExtension?(Fraction(Integer),SparseUnivariatePolynomial?(Fraction(Integer)),?^12+(-(?^10))+?^8+(-(?^6))+?^4+(-(?^2))+1)

fricas

s5 := (e5 - 1/e5)/(2*im2)

$\label{eq22}-{{\frac{1}{2}}\ {{?}^{11}}}+{{\frac{1}{2}}\ {{?}^{3}}}$

(22)

Type: SimpleAlgebraicExtension?(Fraction(Integer),SparseUnivariatePolynomial?(Fraction(Integer)),?^12+(-(?^10))+?^8+(-(?^6))+?^4+(-(?^2))+1)

Using this we can compute the expression:

fricas

im2*(5*s5 - 5*s4 + 5*s3 - 5*s2) - 5*c5 + 5*c4 - 5*c3 + 5*c2 - 10*c1 + 6

$\label{eq23}1$

(23)

Type: SimpleAlgebraicExtension?(Fraction(Integer),SparseUnivariatePolynomial?(Fraction(Integer)),?^12+(-(?^10))+?^8+(-(?^6))+?^4+(-(?^2))+1)