Triangle heptagonal

Les côtés du triangle heptagonal a < b < c coïncident, par définition, avec le côté de l'heptagone régulier, sa diagonale courte et sa diagonale longue. Ces trois longueurs vérifient[3]^{:Lemma 1}

${\displaystyle a^{2}=c(c-b),\ b^{2}=a(c+a),\ c^{2}=b(a+b),\ {\frac {1}{a}}={\frac {1}{b}}+{\frac {1}{c}}}$

(la dernière est connue sous le nom d'équation optique (en)[2]^{:p. 13}) et donc

${\displaystyle ab+ac=bc,}$

et[3]^{:Coro. 2}

${\displaystyle b^{3}+2b^{2}c-bc^{2}-c^{3}=0,}$

${\displaystyle c^{3}-2c^{2}a-ca^{2}+a^{3}=0,}$

${\displaystyle a^{3}-2a^{2}b-ab^{2}+b^{3}=0.}$

Ainsi, les rapports –b/c, c/a, et a/b sont les racines de l'équation cubique

${\displaystyle t^{3}-2t^{2}-t+1=0.}$

Il n'existe aucune expression algébrique réelle pour les solutions de cette équation, car c'est un exemple de casus irreducibilis. On a cependant les approximations

${\displaystyle b\approx 1,80193\cdot a,\qquad c\approx 2,24698\cdot a.}$

On a aussi[4] - [5]

${\displaystyle {\frac {a^{2}}{bc}},\quad -{\frac {b^{2}}{ca}},\quad -{\frac {c^{2}}{ab}}}$

qui vérifient l'équation cubique

${\displaystyle t^{3}+4t^{2}+3t-1=0.}$

On a [4]

${\displaystyle {\frac {a^{3}}{bc^{2}}},\quad -{\frac {b^{3}}{ca^{2}}},\quad {\frac {c^{3}}{ab^{2}}}}$

qui vérifient l'équation cubique

${\displaystyle t^{3}-t^{2}-9t+1=0.}$

On a [4]

${\displaystyle {\frac {a^{3}}{b^{2}c}},\quad {\frac {b^{3}}{c^{2}a}},\quad -{\frac {c^{3}}{a^{2}b}}}$

qui vérifient l'équation cubique

${\displaystyle t^{3}+5t^{2}-8t+1=0.}$

On a [2]^{:p. 14}

${\displaystyle b^{2}-a^{2}=ac,\ c^{2}-b^{2}=ab,\ a^{2}-c^{2}=-bc,}$

et[2]^{:p. 15}

${\displaystyle {\frac {b^{2}}{a^{2}}}+{\frac {c^{2}}{b^{2}}}+{\frac {a^{2}}{c^{2}}}=5.}$

On a aussi[4]

${\displaystyle ab-bc+ca=0,}$

${\displaystyle a^{3}b-b^{3}c+c^{3}a=0,}$

${\displaystyle a^{4}b+b^{4}c-c^{4}a=0,}$

${\displaystyle a^{11}b^{3}-b^{11}c^{3}+c^{11}a^{3}=0.}$

Il n'existe aucun autre couple d'entiers strictement positifs (m, n), m, n > 0, m, n < 2000 tel que

${\displaystyle a^{m}b^{n}\pm b^{m}c^{n}\pm c^{m}a^{n}=0.}$

Les hauteurs h_a, h_b et h_c vérifient[2]^{:p. 13-14}

${\displaystyle h_{a}=h_{b}+h_{c}}$

et

${\displaystyle h_{a}^{2}+h_{b}^{2}+h_{c}^{2}={\frac {a^{2}+b^{2}+c^{2}}{2}}}$[2]^{:p. 14}.

La hauteur pour le côté b (d'angle opposé B) est la moitié de la bissectrice interne w_A de A[2]^{:p. 19} :

${\displaystyle 2h_{b}=w_{A}.}$

Ici, l'angle A est le plus petit angle, et B le second plus petit.

Les longueurs des bissectrices internes w_A, w_B et w_C (bissectrices des angles A, B et C respectivement) vérifient[2]^{:p. 16} :

${\displaystyle w_{A}=b+c,\ w_{B}=c-a,w_{C}=b-a.}$

On note R le rayon du cercle circonscrit au triangle heptagonal. Son aire vaut alors[6] :

${\displaystyle A={\frac {\sqrt {7}}{4}}R^{2}.}$

On a aussi[2]^{:p. 12,15} - [7]

${\displaystyle a^{2}+b^{2}+c^{2}=7R^{2},\quad a^{4}+b^{4}+c^{4}=21R^{4},\quad a^{6}+b^{6}+c^{6}=70R^{6},}$

${\displaystyle {\frac {1}{a^{2}}}+{\frac {1}{b^{2}}}+{\frac {1}{c^{2}}}={\frac {2}{R^{2}}},\quad {\frac {1}{a^{4}}}+{\frac {1}{b^{4}}}+{\frac {1}{c^{4}}}={\frac {2}{R^{4}}},\quad {\frac {1}{a^{6}}}+{\frac {1}{b^{6}}}+{\frac {1}{c^{6}}}={\frac {17}{7R^{6}}}.}$

De façon générale, pour tout entier n,

${\displaystyle a^{2n}+b^{2n}+c^{2n}=g(n)(2R)^{2n}}$

avec

${\displaystyle g(-1)=8,\quad g(0)=3,\quad g(1)=7}$

et

${\displaystyle g(n)=7g(n-1)-14g(n-2)+7g(n-3),}$

on a[7]

${\displaystyle 2b^{2}-a^{2}={\sqrt {7}}bR,\quad 2c^{2}-b^{2}={\sqrt {7}}cR,\quad 2a^{2}-c^{2}=-{\sqrt {7}}aR.}$

On a aussi[4]

${\displaystyle a^{3}c+b^{3}a-c^{3}b=-7R^{4},}$

${\displaystyle a^{4}c-b^{4}a+c^{4}b=7{\sqrt {7}}R^{5},}$

${\displaystyle a^{11}c^{3}+b^{11}a^{3}-c^{11}b^{3}=-7^{3}17R^{14}.}$

Le rapport r/R entre le rayon du cercle inscrit et celui du cercle circonscrit est la racine positive de l'équation cubique[6]

${\displaystyle 8x^{3}+28x^{2}+14x-7=0.}$

Le rayon du cercle exinscrit au côté a est égal au rayon du cercle d'Euler du triangle heptagonal[2]^{:p. 15}.

Les racines de l'équation cubique[4]

${\displaystyle x^{3}-{\frac {\sqrt {7}}{2}}x^{2}+{\frac {\sqrt {7}}{8}}=0}$

sont ${\displaystyle \sin \left({\frac {2\pi }{7}}\right),\sin \left({\frac {4\pi }{7}}\right),\sin \left({\frac {8\pi }{7}}\right).}$

Les racines de l'équation cubique[2]^{:p. 14}

${\displaystyle 64y^{3}-112y^{2}+56y-7=0}$

sont ${\displaystyle \sin ^{2}\left({\frac {\pi }{7}}\right),\sin ^{2}\left({\frac {2\pi }{7}}\right),\sin ^{2}\left({\frac {4\pi }{7}}\right).}$

On a aussi[7] :

${\displaystyle \sin A-\sin B-\sin C=-{\frac {\sqrt {7}}{2}},}$

${\displaystyle \sin A\sin B-\sin B\sin C+\sin C\sin A=0,}$

${\displaystyle \sin A\sin B\sin C={\frac {\sqrt {7}}{8}}.}$

Pour un entier n, on pose S(n) = (-sin A)ⁿ + sinⁿ B + sinⁿ C. On a alors

n	0	1	2	3	4	5	6	7	8	9	10	11	12	13	14	15	16	17	18	19	20
S(n)	3	${\displaystyle {\tfrac {\sqrt {7}}{2}}}$	${\displaystyle {\tfrac {7}{2^{2}}}}$	${\displaystyle {\tfrac {\sqrt {7}}{2}}}$	${\displaystyle {\tfrac {7\cdot 3}{2^{4}}}}$	${\displaystyle {\tfrac {7{\sqrt {7}}}{2^{4}}}}$	${\displaystyle {\tfrac {7\cdot 5}{2^{5}}}}$	${\displaystyle {\tfrac {7^{2}{\sqrt {7}}}{2^{7}}}}$	${\displaystyle {\tfrac {7^{2}\cdot 5}{2^{8}}}}$	${\displaystyle {\tfrac {7\cdot 25{\sqrt {7}}}{2^{9}}}}$	${\displaystyle {\tfrac {7^{2}\cdot 9}{2^{9}}}}$	${\displaystyle {\tfrac {7^{2}\cdot 13{\sqrt {7}}}{2^{11}}}}$	${\displaystyle {\tfrac {7^{2}\cdot 33}{2^{11}}}}$	${\displaystyle {\tfrac {7^{2}\cdot 3{\sqrt {7}}}{2^{9}}}}$	${\displaystyle {\tfrac {7^{4}\cdot 5}{2^{14}}}}$	${\displaystyle {\tfrac {7^{2}\cdot 179{\sqrt {7}}}{2^{15}}}}$	${\displaystyle {\tfrac {7^{3}\cdot 131}{2^{16}}}}$	${\displaystyle {\tfrac {7^{3}\cdot 3{\sqrt {7}}}{2^{12}}}}$	${\displaystyle {\tfrac {7^{3}\cdot 493}{2^{18}}}}$	${\displaystyle {\tfrac {7^{3}\cdot 181{\sqrt {7}}}{2^{18}}}}$	${\displaystyle {\tfrac {7^{5}\cdot 19}{2^{19}}}}$
S(-n)	3	0	23	${\displaystyle -{\tfrac {2^{3}\cdot 3{\sqrt {7}}}{7}}}$	25	${\displaystyle -{\tfrac {2^{5}\cdot 5{\sqrt {7}}}{7}}}$	${\displaystyle {\tfrac {2^{6}\cdot 17}{7}}}$	${\displaystyle -2^{7}{\sqrt {7}}}$	${\displaystyle {\tfrac {2^{9}\cdot 11}{7}}}$	${\displaystyle -{\tfrac {2^{10}\cdot 33{\sqrt {7}}}{7^{2}}}}$	${\displaystyle {\tfrac {2^{10}\cdot 29}{7}}}$	${\displaystyle -{\tfrac {2^{14}\cdot 11{\sqrt {7}}}{7^{2}}}}$	${\displaystyle {\tfrac {2^{12}\cdot 269}{7^{2}}}}$	${\displaystyle -{\tfrac {2^{13}\cdot 117{\sqrt {7}}}{7^{2}}}}$	${\displaystyle {\tfrac {2^{14}\cdot 51}{7}}}$	${\displaystyle -{\tfrac {2^{21}\cdot 17{\sqrt {7}}}{7^{3}}}}$	${\displaystyle {\tfrac {2^{17}\cdot 237}{7^{2}}}}$	${\displaystyle -{\tfrac {2^{17}\cdot 1445{\sqrt {7}}}{7^{3}}}}$	${\displaystyle {\tfrac {2^{19}\cdot 2203}{7^{3}}}}$	${\displaystyle -{\tfrac {2^{19}\cdot 1919{\sqrt {7}}}{7^{3}}}}$	${\displaystyle {\tfrac {2^{20}\cdot 5851}{7^{3}}}}$

Les cosinus aux angles cos A, cos B, cos C sont les racines de l'équation cubique :

${\displaystyle x^{3}+{\frac {1}{2}}x^{2}-{\frac {1}{2}}x-{\frac {1}{8}}=0.}$

Pour un entier n, on pose C(n) = (-cos A)ⁿ + cosⁿ B + cosⁿ C. On a alors

n	0	1	2	3	4	5	6	7	8	9	10
C(n)	3	${\displaystyle -{\frac {1}{2}}}$	${\displaystyle {\frac {5}{4}}}$	${\displaystyle -{\frac {1}{2}}}$	${\displaystyle {\frac {13}{16}}}$	${\displaystyle -{\frac {1}{2}}}$	${\displaystyle {\frac {19}{32}}}$	${\displaystyle -{\frac {57}{128}}}$	${\displaystyle {\frac {117}{256}}}$	${\displaystyle -{\frac {193}{512}}}$	${\displaystyle {\frac {185}{512}}}$
C(-n)	3	-4	24	-88	416	-1824	8256	-36992	166400	-747520	3359744

Les tangentes aux angles tan A, tan B, tan C sont les racines de l'équation cubique :

${\displaystyle x^{3}+{\sqrt {7}}x^{2}-7x+{\sqrt {7}}=0.}$

Les carrés des tangentes aux angles tan²A, tan² B, tan² C sont les racines de l'équation cubique :

${\displaystyle x^{3}-21x^{2}+35x-7=0.}$

Pour un entier n, on pose T(n) = tanⁿ A + tanⁿ B + tanⁿ C. On a alors

n	0	1	2	3	4	5	6	7	8	9	10
T(n)	3	${\displaystyle -{\sqrt {7}}}$	${\displaystyle 7\cdot 3}$	${\displaystyle -31{\sqrt {7}}}$	${\displaystyle 7\cdot 53}$	${\displaystyle -7\cdot 87{\sqrt {7}}}$	${\displaystyle 7\cdot 1011}$	${\displaystyle -7^{2}\cdot 239{\sqrt {7}}}$	${\displaystyle 7^{2}\cdot 2771}$	${\displaystyle -7\cdot 32119{\sqrt {7}}}$	${\displaystyle 7^{2}\cdot 53189}$
T(-n)	3	${\displaystyle {\sqrt {7}}}$	5	${\displaystyle {\frac {25{\sqrt {7}}}{7}}}$	19	${\displaystyle {\frac {103{\sqrt {7}}}{7}}}$	${\displaystyle {\frac {563}{7}}}$	${\displaystyle 7\cdot 9{\sqrt {7}}}$	${\displaystyle {\frac {2421}{7}}}$	${\displaystyle {\frac {13297{\sqrt {7}}}{7^{2}}}}$	${\displaystyle {\frac {10435}{7}}}$

On a aussi[7] - [9]

${\displaystyle \tan A-4\sin B=-{\sqrt {7}},}$

${\displaystyle \tan B-4\sin C=-{\sqrt {7}},}$

${\displaystyle \tan C+4\sin A=-{\sqrt {7}}.}$

On a aussi[4]

${\displaystyle \cot ^{2}A=1-{\frac {2\tan C}{\sqrt {7}}},}$

${\displaystyle \cot ^{2}B=1-{\frac {2\tan A}{\sqrt {7}}},}$

${\displaystyle \cot ^{2}C=1-{\frac {2\tan B}{\sqrt {7}}}.}$

${\displaystyle \cos A=-{\frac {1}{2}}+{\frac {4}{\sqrt {7}}}\sin ^{3}C,}$

${\displaystyle \cos ^{2}A={\frac {3}{4}}+{\frac {2}{\sqrt {7}}}\sin ^{3}A,}$

${\displaystyle \cot A={\frac {3}{\sqrt {7}}}+{\frac {4}{\sqrt {7}}}\cos B,}$

${\displaystyle \cot ^{2}A=3+{\frac {8}{\sqrt {7}}}\sin A,}$

${\displaystyle \cot A={\sqrt {7}}+{\frac {8}{\sqrt {7}}}\sin ^{2}B,}$

${\displaystyle \csc ^{3}A=-{\frac {6}{\sqrt {7}}}+{\frac {2}{\sqrt {7}}}\tan ^{2}C,}$

${\displaystyle \sec A=2+4\cos C,}$

${\displaystyle \sec A=6-8\sin ^{2}B,}$

${\displaystyle \sec A=4-{\frac {16}{\sqrt {7}}}\sin ^{3}B,}$

${\displaystyle \sin ^{2}A={\frac {1}{2}}+{\frac {1}{2}}\cos B,}$

${\displaystyle \sin ^{3}A=-{\frac {\sqrt {7}}{8}}+{\frac {\sqrt {7}}{4}}\cos B,}$

On a aussi[10]

${\displaystyle \sin ^{3}B\sin C-\sin ^{3}C\sin A-\sin ^{3}A\sin B=0,}$

${\displaystyle \sin B\sin ^{3}C-\sin C\sin ^{3}A-\sin A\sin ^{3}B={\frac {7}{2^{4}}},}$

${\displaystyle \sin ^{4}B\sin C-\sin ^{4}C\sin A+\sin ^{4}A\sin B=0,}$

${\displaystyle \sin B\sin ^{4}C+\sin C\sin ^{4}A-\sin A\sin ^{4}B={\frac {7{\sqrt {7}}}{2^{5}}},}$

${\displaystyle \sin ^{11}B\sin ^{3}C-\sin ^{11}C\sin ^{3}A-\sin ^{11}A\sin ^{3}B=0,}$

${\displaystyle \sin ^{3}B\sin ^{11}C-\sin ^{3}C\sin ^{11}A-\sin ^{3}A\sin ^{11}B={\frac {7^{3}\cdot 17}{2^{14}}}.}$

On peut également obtenir des identités similaires à celles découvertes par Srinivasa Ramanujan[7] - [11]

${\displaystyle {\sqrt[{3}]{2\sin \left({\frac {2\pi }{7}}\right)}}+{\sqrt[{3}]{2\sin \left({\frac {4\pi }{7}}\right)}}+{\sqrt[{3}]{2\sin \left({\frac {8\pi }{7}}\right)}}=\left(-{\sqrt[{18}]{7}}\right){\sqrt[{3}]{-{\sqrt[{3}]{7}}+6+3\left({\sqrt[{3}]{5-3{\sqrt[{3}]{7}}}}+{\sqrt[{3}]{4-3{\sqrt[{3}]{7}}}}\right)}}}$

${\displaystyle {\frac {1}{\sqrt[{3}]{2\sin \left({\frac {2\pi }{7}}\right)}}}+{\frac {1}{\sqrt[{3}]{2\sin \left({\frac {4\pi }{7}}\right)}}}+{\frac {1}{\sqrt[{3}]{2\sin \left({\frac {8\pi }{7}}\right)}}}=\left(-{\frac {1}{\sqrt[{18}]{7}}}\right){\sqrt[{3}]{6+3\left({\sqrt[{3}]{5-3{\sqrt[{3}]{7}}}}+{\sqrt[{3}]{4-3{\sqrt[{3}]{7}}}}\right)}}}$

${\displaystyle {\sqrt[{3}]{4\sin ^{2}\left({\frac {2\pi }{7}}\right)}}+{\sqrt[{3}]{4\sin ^{2}\left({\frac {4\pi }{7}}\right)}}+{\sqrt[{3}]{4\sin ^{2}\left({\frac {8\pi }{7}}\right)}}=\left({\sqrt[{18}]{49}}\right){\sqrt[{3}]{{\sqrt[{3}]{49}}+6+3\left({\sqrt[{3}]{12+3({\sqrt[{3}]{49}}+2{\sqrt[{3}]{7}})}}+{\sqrt[{3}]{11+3({\sqrt[{3}]{49}}+2{\sqrt[{3}]{7}})}}\right)}}}$

${\displaystyle {\frac {1}{\sqrt[{3}]{4\sin ^{2}\left({\frac {2\pi }{7}}\right)}}}+{\frac {1}{\sqrt[{3}]{4\sin ^{2}\left({\frac {4\pi }{7}}\right)}}}+{\frac {1}{\sqrt[{3}]{4\sin ^{2}\left({\frac {8\pi }{7}}\right)}}}=\left({\frac {1}{\sqrt[{18}]{49}}}\right){\sqrt[{3}]{2{\sqrt[{3}]{7}}+6+3\left({\sqrt[{3}]{12+3({\sqrt[{3}]{49}}+2{\sqrt[{3}]{7}})}}+{\sqrt[{3}]{11+3({\sqrt[{3}]{49}}+2{\sqrt[{3}]{7}})}}\right)}}}$

${\displaystyle {\sqrt[{3}]{2\cos \left({\frac {2\pi }{7}}\right)}}+{\sqrt[{3}]{2\cos \left({\frac {4\pi }{7}}\right)}}+{\sqrt[{3}]{2\cos \left({\frac {8\pi }{7}}\right)}}={\sqrt[{3}]{5-3{\sqrt[{3}]{7}}}}}$

${\displaystyle {\frac {1}{\sqrt[{3}]{2\cos \left({\frac {2\pi }{7}}\right)}}}+{\frac {1}{\sqrt[{3}]{2\cos \left({\frac {4\pi }{7}}\right)}}}+{\frac {1}{\sqrt[{3}]{2\cos \left({\frac {8\pi }{7}}\right)}}}={\sqrt[{3}]{4-3{\sqrt[{3}]{7}}}}}$

${\displaystyle {\sqrt[{3}]{4\cos ^{2}\left({\frac {2\pi }{7}}\right)}}+{\sqrt[{3}]{4\cos ^{2}\left({\frac {4\pi }{7}}\right)}}+{\sqrt[{3}]{4\cos ^{2}\left({\frac {8\pi }{7}}\right)}}={\sqrt[{3}]{11+3(2{\sqrt[{3}]{7}}+{\sqrt[{3}]{49}})}}}$

${\displaystyle {\frac {1}{\sqrt[{3}]{4\cos ^{2}\left({\frac {2\pi }{7}}\right)}}}+{\frac {1}{\sqrt[{3}]{4\cos ^{2}\left({\frac {4\pi }{7}}\right)}}}+{\frac {1}{\sqrt[{3}]{4\cos ^{2}\left({\frac {8\pi }{7}}\right)}}}={\sqrt[{3}]{12+3(2{\sqrt[{3}]{7}}+{\sqrt[{3}]{49}})}}}$

${\displaystyle {\sqrt[{3}]{\tan \left({\frac {2\pi }{7}}\right)}}+{\sqrt[{3}]{\tan \left({\frac {4\pi }{7}}\right)}}+{\sqrt[{3}]{\tan \left({\frac {8\pi }{7}}\right)}}=\left(-{\sqrt[{18}]{7}}\right){\sqrt[{3}]{{\sqrt[{3}]{7}}+6+3\left({\sqrt[{3}]{5+3({\sqrt[{3}]{7}}-{\sqrt[{3}]{49}})}}+{\sqrt[{3}]{-3+3({\sqrt[{3}]{7}}-{\sqrt[{3}]{49}})}}\right)}}}$

${\displaystyle {\frac {1}{\sqrt[{3}]{\tan \left({\frac {2\pi }{7}}\right)}}}+{\frac {1}{\sqrt[{3}]{\tan \left({\frac {4\pi }{7}}\right)}}}+{\frac {1}{\sqrt[{3}]{\tan \left({\frac {8\pi }{7}}\right)}}}=\left(-{\frac {1}{\sqrt[{18}]{7}}}\right){\sqrt[{3}]{-{\sqrt[{3}]{49}}+6+3\left({\sqrt[{3}]{5+3({\sqrt[{3}]{7}}-{\sqrt[{3}]{49}})}}+{\sqrt[{3}]{-3+3({\sqrt[{3}]{7}}-{\sqrt[{3}]{49}})}}\right)}}}$

${\displaystyle {\sqrt[{3}]{\tan ^{2}\left({\frac {2\pi }{7}}\right)}}+{\sqrt[{3}]{\tan ^{2}\left({\frac {4\pi }{7}}\right)}}+{\sqrt[{3}]{\tan ^{2}\left({\frac {8\pi }{7}}\right)}}=\left({\sqrt[{18}]{49}}\right){\sqrt[{3}]{3{\sqrt[{3}]{49}}+6+3\left({\sqrt[{3}]{89+3(3{\sqrt[{3}]{49}}+5{\sqrt[{3}]{7}})}}+{\sqrt[{3}]{25+3(3{\sqrt[{3}]{49}}+5{\sqrt[{3}]{7}})}}\right)}}}$

${\displaystyle {\frac {1}{\sqrt[{3}]{\tan ^{2}\left({\frac {2\pi }{7}}\right)}}}+{\frac {1}{\sqrt[{3}]{\tan ^{2}\left({\frac {4\pi }{7}}\right)}}}+{\frac {1}{\sqrt[{3}]{\tan ^{2}\left({\frac {8\pi }{7}}\right)}}}=\left({\frac {1}{\sqrt[{18}]{49}}}\right){\sqrt[{3}]{5{\sqrt[{3}]{7}}+6+3\left({\sqrt[{3}]{89+3(3{\sqrt[{3}]{49}}+5{\sqrt[{3}]{7}})}}+{\sqrt[{3}]{25+3(3{\sqrt[{3}]{49}}+5{\sqrt[{3}]{7}})}}\right)}}}$

On a aussi[10]

${\displaystyle {\sqrt[{3}]{\cos \left({\frac {2\pi }{7}}\right)/\cos \left({\frac {4\pi }{7}}\right)}}+{\sqrt[{3}]{\cos \left({\frac {4\pi }{7}}\right)/\cos \left({\frac {8\pi }{7}}\right)}}+{\sqrt[{3}]{\cos \left({\frac {8\pi }{7}}\right)/\cos \left({\frac {2\pi }{7}}\right)}}=-{\sqrt[{3}]{7}}.}$

${\displaystyle {\sqrt[{3}]{\cos \left({\frac {4\pi }{7}}\right)/\cos \left({\frac {2\pi }{7}}\right)}}+{\sqrt[{3}]{\cos \left({\frac {8\pi }{7}}\right)/\cos \left({\frac {4\pi }{7}}\right)}}+{\sqrt[{3}]{\cos \left({\frac {2\pi }{7}}\right)/\cos \left({\frac {8\pi }{7}}\right)}}=0.}$

${\displaystyle {\sqrt[{3}]{2\sin \left({\frac {2\pi }{7}}\right)}}+{\sqrt[{3}]{2\sin \left({\frac {4\pi }{7}}\right)}}+{\sqrt[{3}]{2\sin \left({\frac {8\pi }{7}}\right)}}=\left(-{\sqrt[{18}]{7}}\right){\sqrt[{3}]{-{\sqrt[{3}]{7}}+6+3\left({\sqrt[{3}]{5-3{\sqrt[{3}]{7}}}}+{\sqrt[{3}]{4-3{\sqrt[{3}]{7}}}}\right)}}}$

${\displaystyle {\sqrt[{3}]{\cos ^{4}\left({\frac {4\pi }{7}}\right)/\cos \left({\frac {2\pi }{7}}\right)}}+{\sqrt[{3}]{\cos ^{4}\left({\frac {8\pi }{7}}\right)/\cos \left({\frac {4\pi }{7}}\right)}}+{\sqrt[{3}]{\cos ^{4}\left({\frac {2\pi }{7}}\right)/\cos \left({\frac {8\pi }{7}}\right)}}=-{\sqrt[{3}]{49}}/2.}$

${\displaystyle {\sqrt[{3}]{\cos ^{5}\left({\frac {2\pi }{7}}\right)/\cos ^{2}\left({\frac {4\pi }{7}}\right)}}+{\sqrt[{3}]{\cos ^{5}\left({\frac {4\pi }{7}}\right)/\cos ^{2}\left({\frac {8\pi }{7}}\right)}}+{\sqrt[{3}]{\cos ^{5}\left({\frac {8\pi }{7}}\right)/\cos ^{2}\left({\frac {2\pi }{7}}\right)}}=0.}$

${\displaystyle {\sqrt[{3}]{\cos ^{5}\left({\frac {4\pi }{7}}\right)/\cos ^{2}\left({\frac {2\pi }{7}}\right)}}+{\sqrt[{3}]{\cos ^{5}\left({\frac {8\pi }{7}}\right)/\cos ^{2}\left({\frac {4\pi }{7}}\right)}}+{\sqrt[{3}]{\cos ^{5}\left({\frac {2\pi }{7}}\right)/\cos ^{2}\left({\frac {8\pi }{7}}\right)}}=-3{\sqrt[{3}]{7}}/2.}$

${\displaystyle {\sqrt[{3}]{\cos ^{14}\left({\frac {2\pi }{7}}\right)/\cos ^{5}\left({\frac {4\pi }{7}}\right)}}+{\sqrt[{3}]{\cos ^{14}\left({\frac {4\pi }{7}}\right)/\cos ^{5}\left({\frac {8\pi }{7}}\right)}}+{\sqrt[{3}]{\cos ^{14}\left({\frac {8\pi }{7}}\right)/\cos ^{5}\left({\frac {2\pi }{7}}\right)}}=0.}$

${\displaystyle {\sqrt[{3}]{\cos ^{14}\left({\frac {4\pi }{7}}\right)/\cos ^{5}\left({\frac {2\pi }{7}}\right)}}+{\sqrt[{3}]{\cos ^{14}\left({\frac {8\pi }{7}}\right)/\cos ^{5}\left({\frac {4\pi }{7}}\right)}}+{\sqrt[{3}]{\cos ^{14}\left({\frac {2\pi }{7}}\right)/\cos ^{5}\left({\frac {8\pi }{7}}\right)}}=-61{\sqrt[{3}]{7}}/8.}$