Evaluation of $sin frac{pi}{7}cdot sin frac{2pi}{7}cdot sin frac{3pi}{7}$












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Evaluation of $$sin frac{pi}{7}cdot sin frac{2pi}{7}cdot sin frac{3pi}{7} = $$




$bf{My; Try::}$ I have solved Using Direct formula::



$$sin frac{pi}{n}cdot sin frac{2pi}{n}cdot......sin frac{(n-1)pi}{n} = frac{n}{2^{n-1}}$$



Now Put $n=7;,$ We get
$$sin frac{pi}{7}cdot sin frac{2pi}{7}cdot sin frac{3pi}{7}cdot sin frac{4pi}{7}cdot sin frac{5pi}{7}cdot sin frac{6pi}{7}=frac{7}{2^{7-1}}$$



So $$sin frac{pi}{7}cdot sin frac{2pi}{7}cdot sin frac{3pi}{7} =frac{sqrt{7}}{8}$$



Now my question is how can we solve it without using Direct Formula, Help me



Thanks










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  • $begingroup$
    Related : math.stackexchange.com/questions/818749/… and math.stackexchange.com/questions/1745060/…
    $endgroup$
    – lab bhattacharjee
    May 14 '16 at 16:09
















6












$begingroup$



Evaluation of $$sin frac{pi}{7}cdot sin frac{2pi}{7}cdot sin frac{3pi}{7} = $$




$bf{My; Try::}$ I have solved Using Direct formula::



$$sin frac{pi}{n}cdot sin frac{2pi}{n}cdot......sin frac{(n-1)pi}{n} = frac{n}{2^{n-1}}$$



Now Put $n=7;,$ We get
$$sin frac{pi}{7}cdot sin frac{2pi}{7}cdot sin frac{3pi}{7}cdot sin frac{4pi}{7}cdot sin frac{5pi}{7}cdot sin frac{6pi}{7}=frac{7}{2^{7-1}}$$



So $$sin frac{pi}{7}cdot sin frac{2pi}{7}cdot sin frac{3pi}{7} =frac{sqrt{7}}{8}$$



Now my question is how can we solve it without using Direct Formula, Help me



Thanks










share|cite|improve this question











$endgroup$












  • $begingroup$
    Related : math.stackexchange.com/questions/818749/… and math.stackexchange.com/questions/1745060/…
    $endgroup$
    – lab bhattacharjee
    May 14 '16 at 16:09














6












6








6


4



$begingroup$



Evaluation of $$sin frac{pi}{7}cdot sin frac{2pi}{7}cdot sin frac{3pi}{7} = $$




$bf{My; Try::}$ I have solved Using Direct formula::



$$sin frac{pi}{n}cdot sin frac{2pi}{n}cdot......sin frac{(n-1)pi}{n} = frac{n}{2^{n-1}}$$



Now Put $n=7;,$ We get
$$sin frac{pi}{7}cdot sin frac{2pi}{7}cdot sin frac{3pi}{7}cdot sin frac{4pi}{7}cdot sin frac{5pi}{7}cdot sin frac{6pi}{7}=frac{7}{2^{7-1}}$$



So $$sin frac{pi}{7}cdot sin frac{2pi}{7}cdot sin frac{3pi}{7} =frac{sqrt{7}}{8}$$



Now my question is how can we solve it without using Direct Formula, Help me



Thanks










share|cite|improve this question











$endgroup$





Evaluation of $$sin frac{pi}{7}cdot sin frac{2pi}{7}cdot sin frac{3pi}{7} = $$




$bf{My; Try::}$ I have solved Using Direct formula::



$$sin frac{pi}{n}cdot sin frac{2pi}{n}cdot......sin frac{(n-1)pi}{n} = frac{n}{2^{n-1}}$$



Now Put $n=7;,$ We get
$$sin frac{pi}{7}cdot sin frac{2pi}{7}cdot sin frac{3pi}{7}cdot sin frac{4pi}{7}cdot sin frac{5pi}{7}cdot sin frac{6pi}{7}=frac{7}{2^{7-1}}$$



So $$sin frac{pi}{7}cdot sin frac{2pi}{7}cdot sin frac{3pi}{7} =frac{sqrt{7}}{8}$$



Now my question is how can we solve it without using Direct Formula, Help me



Thanks







trigonometry






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edited May 14 '16 at 8:21









Jean-Claude Arbaut

14.7k63464




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asked May 14 '16 at 7:56









juantheronjuantheron

34.2k1147142




34.2k1147142












  • $begingroup$
    Related : math.stackexchange.com/questions/818749/… and math.stackexchange.com/questions/1745060/…
    $endgroup$
    – lab bhattacharjee
    May 14 '16 at 16:09


















  • $begingroup$
    Related : math.stackexchange.com/questions/818749/… and math.stackexchange.com/questions/1745060/…
    $endgroup$
    – lab bhattacharjee
    May 14 '16 at 16:09
















$begingroup$
Related : math.stackexchange.com/questions/818749/… and math.stackexchange.com/questions/1745060/…
$endgroup$
– lab bhattacharjee
May 14 '16 at 16:09




$begingroup$
Related : math.stackexchange.com/questions/818749/… and math.stackexchange.com/questions/1745060/…
$endgroup$
– lab bhattacharjee
May 14 '16 at 16:09










4 Answers
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$begingroup$

$$
begin{align}
prod_{k=1}^3sinleft(frac{kpi}7right)^2
&=prod_{k=1}^6sinleft(frac{kpi}7right)tag{1}\
&=-frac1{64}prod_{k=1}^6left(e^{ikpi/7}-e^{-ikpi/7}right)tag{2}\
&=frac1{64}prod_{k=1}^6left(1-e^{-i2kpi/7}right)tag{3}\
&=frac1{64}lim_{zto1}prod_{k=1}^6left(z-e^{-i2kpi/7}right)tag{4}\
&=frac1{64}lim_{zto1}frac{z^7-1}{z-1}tag{5}\[6pt]
&=frac7{64}tag{6}
end{align}
$$
Explanation:

$(1)$: $sin(x)=sin(pi-x)$

$(2)$: $sin(x)=frac{e^{ix}-e^{-ix}}{2i}$

$(3)$: pull $prodlimits_{k=1}^6e^{ikpi/7}=-1$ out of the product

$(4)$: $1=limlimits_{zto1}z$

$(5)$: $prodlimits_{k=1}^6left(z-e^{-i2kpi/7}right)=frac{z^7-1}{z-1}$

$(6)$: evaluate limit



Therefore, taking the square root of $(6)$, we get
$$
prod_{k=1}^3sinleft(frac{kpi}7right)=frac{sqrt7}8tag{7}
$$






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    2












    $begingroup$

    Using $2sin asin b=cos(a-b)-cos(a+b)$ and $2sin acos b=sin(a+b)+sin(a-b)$, write



    $$sin frac{pi}7cdotsin frac{2pi}7cdot sin frac{3pi}7 = frac12left(cosfrac{pi}7-cosfrac{3pi}7right)sinfrac{3pi}7=frac14left(sinfrac{4pi}7+sinfrac{2pi}7-sinfrac{pi}7right)\=frac14left(sinfrac{2pi}7+sinfrac{4pi}7+sinfrac{8pi}7right)$$



    Then have a look at this question: Trigo Problem : Find the value of $sinfrac{2pi}{7}+sinfrac{4pi}{7}+sinfrac{8pi}{7}$






    share|cite|improve this answer











    $endgroup$





















      2












      $begingroup$

      This is a Gauss sum in disguise, but it can be computed through elementary geometry, without resorting to algebraic number theory. Let us consider a regular heptagon $P_1 P_2 P_3 P_4 P_5 P_6 P_7$ with center $O$ and take $A=P_1, B=P_5, C=P_6$. The angles of $ABC$ are $frac{pi}{7},frac{2pi}{7},frac{4pi}{7}$. By Euler's formula and the sine theorem



      $$ [ABC] = frac{abc}{4R} = 2R^2 sin(A)sin(B)sin(C) $$
      but we also have
      $$[ABC] = [AOB]+[BOC]-[AOC] = frac{R^2}{2}left(sinfrac{4pi}{7}+sinfrac{2pi}{7}-sinfrac{6pi}{7}right) $$
      hence the claim is equivalent to:
      $$ sinfrac{2pi}{7}+sinfrac{4pi}{7}+sinfrac{8pi}{7} = frac{1}{2}sqrt{7}. $$
      and since the values of the sine function at $frac{pi}{7},frac{2pi}{7},frac{4pi}{7}$ are positive, the claim is also equivalent to:
      $$ left(1-cosfrac{2pi}{7}right)left(1-cosfrac{4pi}{7}right)left(1-cosfrac{8pi}{7}right)=frac{7}{8}. $$
      Chebyshev polynomials ensure that $cosfrac{2pi}{7},cosfrac{4pi}{7},cosfrac{8pi}{7}$ are algebraic conjugates over $mathbb{Q}$, and they are the roots of $p(x)=8x^3+4x^2-4x-1$. By factoring such polynomial over $mathbb{R}$ it follows that
      $$ p(x) = 8x^3+4x^2-4x-1 = 8left(x-cosfrac{2pi}{7}right)left(x-cosfrac{4pi}{7}right)left(x-cosfrac{8pi}{7}right) $$
      and by evaluating at $x=1$ we get
      $$ 7 = p(1) = 8left(1-cosfrac{2pi}{7}right)left(1-cosfrac{4pi}{7}right)left(1-cosfrac{8pi}{7}right) $$
      QED.






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        Hint;



        Assume
        $$ theta = (npi)/7$$



        $$implies 4theta = npi -3theta$$



        Take the sine on both the sides
        An apply the required manipulations



        You should get a cubic in $sin^2 theta$
        Since we have defined out $theta$, we are quite aware what the roots are going to be



        So the numerical part in the end divided by the coefficient of $x^3$ gives you



        $$ {{[sin frac{pi}{7}cdot sin frac{2pi}{7}cdot sin frac{3pi}{7}}]}^2 = 7/64$$



        That should be it






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          4 Answers
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          4 Answers
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          3












          $begingroup$

          $$
          begin{align}
          prod_{k=1}^3sinleft(frac{kpi}7right)^2
          &=prod_{k=1}^6sinleft(frac{kpi}7right)tag{1}\
          &=-frac1{64}prod_{k=1}^6left(e^{ikpi/7}-e^{-ikpi/7}right)tag{2}\
          &=frac1{64}prod_{k=1}^6left(1-e^{-i2kpi/7}right)tag{3}\
          &=frac1{64}lim_{zto1}prod_{k=1}^6left(z-e^{-i2kpi/7}right)tag{4}\
          &=frac1{64}lim_{zto1}frac{z^7-1}{z-1}tag{5}\[6pt]
          &=frac7{64}tag{6}
          end{align}
          $$
          Explanation:

          $(1)$: $sin(x)=sin(pi-x)$

          $(2)$: $sin(x)=frac{e^{ix}-e^{-ix}}{2i}$

          $(3)$: pull $prodlimits_{k=1}^6e^{ikpi/7}=-1$ out of the product

          $(4)$: $1=limlimits_{zto1}z$

          $(5)$: $prodlimits_{k=1}^6left(z-e^{-i2kpi/7}right)=frac{z^7-1}{z-1}$

          $(6)$: evaluate limit



          Therefore, taking the square root of $(6)$, we get
          $$
          prod_{k=1}^3sinleft(frac{kpi}7right)=frac{sqrt7}8tag{7}
          $$






          share|cite|improve this answer











          $endgroup$


















            3












            $begingroup$

            $$
            begin{align}
            prod_{k=1}^3sinleft(frac{kpi}7right)^2
            &=prod_{k=1}^6sinleft(frac{kpi}7right)tag{1}\
            &=-frac1{64}prod_{k=1}^6left(e^{ikpi/7}-e^{-ikpi/7}right)tag{2}\
            &=frac1{64}prod_{k=1}^6left(1-e^{-i2kpi/7}right)tag{3}\
            &=frac1{64}lim_{zto1}prod_{k=1}^6left(z-e^{-i2kpi/7}right)tag{4}\
            &=frac1{64}lim_{zto1}frac{z^7-1}{z-1}tag{5}\[6pt]
            &=frac7{64}tag{6}
            end{align}
            $$
            Explanation:

            $(1)$: $sin(x)=sin(pi-x)$

            $(2)$: $sin(x)=frac{e^{ix}-e^{-ix}}{2i}$

            $(3)$: pull $prodlimits_{k=1}^6e^{ikpi/7}=-1$ out of the product

            $(4)$: $1=limlimits_{zto1}z$

            $(5)$: $prodlimits_{k=1}^6left(z-e^{-i2kpi/7}right)=frac{z^7-1}{z-1}$

            $(6)$: evaluate limit



            Therefore, taking the square root of $(6)$, we get
            $$
            prod_{k=1}^3sinleft(frac{kpi}7right)=frac{sqrt7}8tag{7}
            $$






            share|cite|improve this answer











            $endgroup$
















              3












              3








              3





              $begingroup$

              $$
              begin{align}
              prod_{k=1}^3sinleft(frac{kpi}7right)^2
              &=prod_{k=1}^6sinleft(frac{kpi}7right)tag{1}\
              &=-frac1{64}prod_{k=1}^6left(e^{ikpi/7}-e^{-ikpi/7}right)tag{2}\
              &=frac1{64}prod_{k=1}^6left(1-e^{-i2kpi/7}right)tag{3}\
              &=frac1{64}lim_{zto1}prod_{k=1}^6left(z-e^{-i2kpi/7}right)tag{4}\
              &=frac1{64}lim_{zto1}frac{z^7-1}{z-1}tag{5}\[6pt]
              &=frac7{64}tag{6}
              end{align}
              $$
              Explanation:

              $(1)$: $sin(x)=sin(pi-x)$

              $(2)$: $sin(x)=frac{e^{ix}-e^{-ix}}{2i}$

              $(3)$: pull $prodlimits_{k=1}^6e^{ikpi/7}=-1$ out of the product

              $(4)$: $1=limlimits_{zto1}z$

              $(5)$: $prodlimits_{k=1}^6left(z-e^{-i2kpi/7}right)=frac{z^7-1}{z-1}$

              $(6)$: evaluate limit



              Therefore, taking the square root of $(6)$, we get
              $$
              prod_{k=1}^3sinleft(frac{kpi}7right)=frac{sqrt7}8tag{7}
              $$






              share|cite|improve this answer











              $endgroup$



              $$
              begin{align}
              prod_{k=1}^3sinleft(frac{kpi}7right)^2
              &=prod_{k=1}^6sinleft(frac{kpi}7right)tag{1}\
              &=-frac1{64}prod_{k=1}^6left(e^{ikpi/7}-e^{-ikpi/7}right)tag{2}\
              &=frac1{64}prod_{k=1}^6left(1-e^{-i2kpi/7}right)tag{3}\
              &=frac1{64}lim_{zto1}prod_{k=1}^6left(z-e^{-i2kpi/7}right)tag{4}\
              &=frac1{64}lim_{zto1}frac{z^7-1}{z-1}tag{5}\[6pt]
              &=frac7{64}tag{6}
              end{align}
              $$
              Explanation:

              $(1)$: $sin(x)=sin(pi-x)$

              $(2)$: $sin(x)=frac{e^{ix}-e^{-ix}}{2i}$

              $(3)$: pull $prodlimits_{k=1}^6e^{ikpi/7}=-1$ out of the product

              $(4)$: $1=limlimits_{zto1}z$

              $(5)$: $prodlimits_{k=1}^6left(z-e^{-i2kpi/7}right)=frac{z^7-1}{z-1}$

              $(6)$: evaluate limit



              Therefore, taking the square root of $(6)$, we get
              $$
              prod_{k=1}^3sinleft(frac{kpi}7right)=frac{sqrt7}8tag{7}
              $$







              share|cite|improve this answer














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              edited May 14 '16 at 11:43

























              answered May 14 '16 at 9:32









              robjohnrobjohn

              267k27308631




              267k27308631























                  2












                  $begingroup$

                  Using $2sin asin b=cos(a-b)-cos(a+b)$ and $2sin acos b=sin(a+b)+sin(a-b)$, write



                  $$sin frac{pi}7cdotsin frac{2pi}7cdot sin frac{3pi}7 = frac12left(cosfrac{pi}7-cosfrac{3pi}7right)sinfrac{3pi}7=frac14left(sinfrac{4pi}7+sinfrac{2pi}7-sinfrac{pi}7right)\=frac14left(sinfrac{2pi}7+sinfrac{4pi}7+sinfrac{8pi}7right)$$



                  Then have a look at this question: Trigo Problem : Find the value of $sinfrac{2pi}{7}+sinfrac{4pi}{7}+sinfrac{8pi}{7}$






                  share|cite|improve this answer











                  $endgroup$


















                    2












                    $begingroup$

                    Using $2sin asin b=cos(a-b)-cos(a+b)$ and $2sin acos b=sin(a+b)+sin(a-b)$, write



                    $$sin frac{pi}7cdotsin frac{2pi}7cdot sin frac{3pi}7 = frac12left(cosfrac{pi}7-cosfrac{3pi}7right)sinfrac{3pi}7=frac14left(sinfrac{4pi}7+sinfrac{2pi}7-sinfrac{pi}7right)\=frac14left(sinfrac{2pi}7+sinfrac{4pi}7+sinfrac{8pi}7right)$$



                    Then have a look at this question: Trigo Problem : Find the value of $sinfrac{2pi}{7}+sinfrac{4pi}{7}+sinfrac{8pi}{7}$






                    share|cite|improve this answer











                    $endgroup$
















                      2












                      2








                      2





                      $begingroup$

                      Using $2sin asin b=cos(a-b)-cos(a+b)$ and $2sin acos b=sin(a+b)+sin(a-b)$, write



                      $$sin frac{pi}7cdotsin frac{2pi}7cdot sin frac{3pi}7 = frac12left(cosfrac{pi}7-cosfrac{3pi}7right)sinfrac{3pi}7=frac14left(sinfrac{4pi}7+sinfrac{2pi}7-sinfrac{pi}7right)\=frac14left(sinfrac{2pi}7+sinfrac{4pi}7+sinfrac{8pi}7right)$$



                      Then have a look at this question: Trigo Problem : Find the value of $sinfrac{2pi}{7}+sinfrac{4pi}{7}+sinfrac{8pi}{7}$






                      share|cite|improve this answer











                      $endgroup$



                      Using $2sin asin b=cos(a-b)-cos(a+b)$ and $2sin acos b=sin(a+b)+sin(a-b)$, write



                      $$sin frac{pi}7cdotsin frac{2pi}7cdot sin frac{3pi}7 = frac12left(cosfrac{pi}7-cosfrac{3pi}7right)sinfrac{3pi}7=frac14left(sinfrac{4pi}7+sinfrac{2pi}7-sinfrac{pi}7right)\=frac14left(sinfrac{2pi}7+sinfrac{4pi}7+sinfrac{8pi}7right)$$



                      Then have a look at this question: Trigo Problem : Find the value of $sinfrac{2pi}{7}+sinfrac{4pi}{7}+sinfrac{8pi}{7}$







                      share|cite|improve this answer














                      share|cite|improve this answer



                      share|cite|improve this answer








                      edited Apr 13 '17 at 12:19









                      Community

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                      answered May 14 '16 at 8:20









                      Jean-Claude ArbautJean-Claude Arbaut

                      14.7k63464




                      14.7k63464























                          2












                          $begingroup$

                          This is a Gauss sum in disguise, but it can be computed through elementary geometry, without resorting to algebraic number theory. Let us consider a regular heptagon $P_1 P_2 P_3 P_4 P_5 P_6 P_7$ with center $O$ and take $A=P_1, B=P_5, C=P_6$. The angles of $ABC$ are $frac{pi}{7},frac{2pi}{7},frac{4pi}{7}$. By Euler's formula and the sine theorem



                          $$ [ABC] = frac{abc}{4R} = 2R^2 sin(A)sin(B)sin(C) $$
                          but we also have
                          $$[ABC] = [AOB]+[BOC]-[AOC] = frac{R^2}{2}left(sinfrac{4pi}{7}+sinfrac{2pi}{7}-sinfrac{6pi}{7}right) $$
                          hence the claim is equivalent to:
                          $$ sinfrac{2pi}{7}+sinfrac{4pi}{7}+sinfrac{8pi}{7} = frac{1}{2}sqrt{7}. $$
                          and since the values of the sine function at $frac{pi}{7},frac{2pi}{7},frac{4pi}{7}$ are positive, the claim is also equivalent to:
                          $$ left(1-cosfrac{2pi}{7}right)left(1-cosfrac{4pi}{7}right)left(1-cosfrac{8pi}{7}right)=frac{7}{8}. $$
                          Chebyshev polynomials ensure that $cosfrac{2pi}{7},cosfrac{4pi}{7},cosfrac{8pi}{7}$ are algebraic conjugates over $mathbb{Q}$, and they are the roots of $p(x)=8x^3+4x^2-4x-1$. By factoring such polynomial over $mathbb{R}$ it follows that
                          $$ p(x) = 8x^3+4x^2-4x-1 = 8left(x-cosfrac{2pi}{7}right)left(x-cosfrac{4pi}{7}right)left(x-cosfrac{8pi}{7}right) $$
                          and by evaluating at $x=1$ we get
                          $$ 7 = p(1) = 8left(1-cosfrac{2pi}{7}right)left(1-cosfrac{4pi}{7}right)left(1-cosfrac{8pi}{7}right) $$
                          QED.






                          share|cite|improve this answer









                          $endgroup$


















                            2












                            $begingroup$

                            This is a Gauss sum in disguise, but it can be computed through elementary geometry, without resorting to algebraic number theory. Let us consider a regular heptagon $P_1 P_2 P_3 P_4 P_5 P_6 P_7$ with center $O$ and take $A=P_1, B=P_5, C=P_6$. The angles of $ABC$ are $frac{pi}{7},frac{2pi}{7},frac{4pi}{7}$. By Euler's formula and the sine theorem



                            $$ [ABC] = frac{abc}{4R} = 2R^2 sin(A)sin(B)sin(C) $$
                            but we also have
                            $$[ABC] = [AOB]+[BOC]-[AOC] = frac{R^2}{2}left(sinfrac{4pi}{7}+sinfrac{2pi}{7}-sinfrac{6pi}{7}right) $$
                            hence the claim is equivalent to:
                            $$ sinfrac{2pi}{7}+sinfrac{4pi}{7}+sinfrac{8pi}{7} = frac{1}{2}sqrt{7}. $$
                            and since the values of the sine function at $frac{pi}{7},frac{2pi}{7},frac{4pi}{7}$ are positive, the claim is also equivalent to:
                            $$ left(1-cosfrac{2pi}{7}right)left(1-cosfrac{4pi}{7}right)left(1-cosfrac{8pi}{7}right)=frac{7}{8}. $$
                            Chebyshev polynomials ensure that $cosfrac{2pi}{7},cosfrac{4pi}{7},cosfrac{8pi}{7}$ are algebraic conjugates over $mathbb{Q}$, and they are the roots of $p(x)=8x^3+4x^2-4x-1$. By factoring such polynomial over $mathbb{R}$ it follows that
                            $$ p(x) = 8x^3+4x^2-4x-1 = 8left(x-cosfrac{2pi}{7}right)left(x-cosfrac{4pi}{7}right)left(x-cosfrac{8pi}{7}right) $$
                            and by evaluating at $x=1$ we get
                            $$ 7 = p(1) = 8left(1-cosfrac{2pi}{7}right)left(1-cosfrac{4pi}{7}right)left(1-cosfrac{8pi}{7}right) $$
                            QED.






                            share|cite|improve this answer









                            $endgroup$
















                              2












                              2








                              2





                              $begingroup$

                              This is a Gauss sum in disguise, but it can be computed through elementary geometry, without resorting to algebraic number theory. Let us consider a regular heptagon $P_1 P_2 P_3 P_4 P_5 P_6 P_7$ with center $O$ and take $A=P_1, B=P_5, C=P_6$. The angles of $ABC$ are $frac{pi}{7},frac{2pi}{7},frac{4pi}{7}$. By Euler's formula and the sine theorem



                              $$ [ABC] = frac{abc}{4R} = 2R^2 sin(A)sin(B)sin(C) $$
                              but we also have
                              $$[ABC] = [AOB]+[BOC]-[AOC] = frac{R^2}{2}left(sinfrac{4pi}{7}+sinfrac{2pi}{7}-sinfrac{6pi}{7}right) $$
                              hence the claim is equivalent to:
                              $$ sinfrac{2pi}{7}+sinfrac{4pi}{7}+sinfrac{8pi}{7} = frac{1}{2}sqrt{7}. $$
                              and since the values of the sine function at $frac{pi}{7},frac{2pi}{7},frac{4pi}{7}$ are positive, the claim is also equivalent to:
                              $$ left(1-cosfrac{2pi}{7}right)left(1-cosfrac{4pi}{7}right)left(1-cosfrac{8pi}{7}right)=frac{7}{8}. $$
                              Chebyshev polynomials ensure that $cosfrac{2pi}{7},cosfrac{4pi}{7},cosfrac{8pi}{7}$ are algebraic conjugates over $mathbb{Q}$, and they are the roots of $p(x)=8x^3+4x^2-4x-1$. By factoring such polynomial over $mathbb{R}$ it follows that
                              $$ p(x) = 8x^3+4x^2-4x-1 = 8left(x-cosfrac{2pi}{7}right)left(x-cosfrac{4pi}{7}right)left(x-cosfrac{8pi}{7}right) $$
                              and by evaluating at $x=1$ we get
                              $$ 7 = p(1) = 8left(1-cosfrac{2pi}{7}right)left(1-cosfrac{4pi}{7}right)left(1-cosfrac{8pi}{7}right) $$
                              QED.






                              share|cite|improve this answer









                              $endgroup$



                              This is a Gauss sum in disguise, but it can be computed through elementary geometry, without resorting to algebraic number theory. Let us consider a regular heptagon $P_1 P_2 P_3 P_4 P_5 P_6 P_7$ with center $O$ and take $A=P_1, B=P_5, C=P_6$. The angles of $ABC$ are $frac{pi}{7},frac{2pi}{7},frac{4pi}{7}$. By Euler's formula and the sine theorem



                              $$ [ABC] = frac{abc}{4R} = 2R^2 sin(A)sin(B)sin(C) $$
                              but we also have
                              $$[ABC] = [AOB]+[BOC]-[AOC] = frac{R^2}{2}left(sinfrac{4pi}{7}+sinfrac{2pi}{7}-sinfrac{6pi}{7}right) $$
                              hence the claim is equivalent to:
                              $$ sinfrac{2pi}{7}+sinfrac{4pi}{7}+sinfrac{8pi}{7} = frac{1}{2}sqrt{7}. $$
                              and since the values of the sine function at $frac{pi}{7},frac{2pi}{7},frac{4pi}{7}$ are positive, the claim is also equivalent to:
                              $$ left(1-cosfrac{2pi}{7}right)left(1-cosfrac{4pi}{7}right)left(1-cosfrac{8pi}{7}right)=frac{7}{8}. $$
                              Chebyshev polynomials ensure that $cosfrac{2pi}{7},cosfrac{4pi}{7},cosfrac{8pi}{7}$ are algebraic conjugates over $mathbb{Q}$, and they are the roots of $p(x)=8x^3+4x^2-4x-1$. By factoring such polynomial over $mathbb{R}$ it follows that
                              $$ p(x) = 8x^3+4x^2-4x-1 = 8left(x-cosfrac{2pi}{7}right)left(x-cosfrac{4pi}{7}right)left(x-cosfrac{8pi}{7}right) $$
                              and by evaluating at $x=1$ we get
                              $$ 7 = p(1) = 8left(1-cosfrac{2pi}{7}right)left(1-cosfrac{4pi}{7}right)left(1-cosfrac{8pi}{7}right) $$
                              QED.







                              share|cite|improve this answer












                              share|cite|improve this answer



                              share|cite|improve this answer










                              answered Dec 11 '18 at 1:58









                              Jack D'AurizioJack D'Aurizio

                              1




                              1























                                  1












                                  $begingroup$

                                  Hint;



                                  Assume
                                  $$ theta = (npi)/7$$



                                  $$implies 4theta = npi -3theta$$



                                  Take the sine on both the sides
                                  An apply the required manipulations



                                  You should get a cubic in $sin^2 theta$
                                  Since we have defined out $theta$, we are quite aware what the roots are going to be



                                  So the numerical part in the end divided by the coefficient of $x^3$ gives you



                                  $$ {{[sin frac{pi}{7}cdot sin frac{2pi}{7}cdot sin frac{3pi}{7}}]}^2 = 7/64$$



                                  That should be it






                                  share|cite|improve this answer









                                  $endgroup$


















                                    1












                                    $begingroup$

                                    Hint;



                                    Assume
                                    $$ theta = (npi)/7$$



                                    $$implies 4theta = npi -3theta$$



                                    Take the sine on both the sides
                                    An apply the required manipulations



                                    You should get a cubic in $sin^2 theta$
                                    Since we have defined out $theta$, we are quite aware what the roots are going to be



                                    So the numerical part in the end divided by the coefficient of $x^3$ gives you



                                    $$ {{[sin frac{pi}{7}cdot sin frac{2pi}{7}cdot sin frac{3pi}{7}}]}^2 = 7/64$$



                                    That should be it






                                    share|cite|improve this answer









                                    $endgroup$
















                                      1












                                      1








                                      1





                                      $begingroup$

                                      Hint;



                                      Assume
                                      $$ theta = (npi)/7$$



                                      $$implies 4theta = npi -3theta$$



                                      Take the sine on both the sides
                                      An apply the required manipulations



                                      You should get a cubic in $sin^2 theta$
                                      Since we have defined out $theta$, we are quite aware what the roots are going to be



                                      So the numerical part in the end divided by the coefficient of $x^3$ gives you



                                      $$ {{[sin frac{pi}{7}cdot sin frac{2pi}{7}cdot sin frac{3pi}{7}}]}^2 = 7/64$$



                                      That should be it






                                      share|cite|improve this answer









                                      $endgroup$



                                      Hint;



                                      Assume
                                      $$ theta = (npi)/7$$



                                      $$implies 4theta = npi -3theta$$



                                      Take the sine on both the sides
                                      An apply the required manipulations



                                      You should get a cubic in $sin^2 theta$
                                      Since we have defined out $theta$, we are quite aware what the roots are going to be



                                      So the numerical part in the end divided by the coefficient of $x^3$ gives you



                                      $$ {{[sin frac{pi}{7}cdot sin frac{2pi}{7}cdot sin frac{3pi}{7}}]}^2 = 7/64$$



                                      That should be it







                                      share|cite|improve this answer












                                      share|cite|improve this answer



                                      share|cite|improve this answer










                                      answered Jul 8 '16 at 9:48









                                      sidt36sidt36

                                      151110




                                      151110






























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