Exact Values of cos(2*k*pi/7) by Literka

It is a well known theorem that a regular heptagon cannot be constructed with a ruler and a compass. This means that the values cos(2*pi/7) cannot be expressed in terms of second degree radicals.

It can be derived from the DeMoivre formulas (see also Polynomials with roots, which are exact trigonometric values) that the polynomial



has roots cos(2*pi/7), cos(4*pi/7), cos(8*pi/7). Knowing formulas of Cardano for roots of cubic equations (see Cubic formulas) it is easy to find these values.
Discriminant of this equation is 49/64. Substituting values to a cubic formula we compute that the value cos(2*pi/7) is equal



Notice that although cos(2*pi/7) is a real value, there is an imaginary number i involved in this expression. It cannot be avoided. This situation is called "Casus Irreducibilis". Literka learned about it from an email received from Satoshi Hoshino.

Let w be a root of third degree of 1, defined by



Using Cardano's formula for cubic equations, we can write two other roots of the equation. The value cos(4*pi/7) is equal



The value cos(8*pi/7) is equal






See a similar page about values of cos(2*k*pi/13).


See the list and descriptions of mathematical pages from Mathematical Countryside.


See other pages of 'Literka' from Mathematical Countryside:
Monotonic subsequences.
A remarkable monotonic property of the gamma function .
An elementary problem can be unsolvable.
Weierstrass Approximation Theorem. Bernstein's Polynomials.
Construction of a regular pentagon.
Roots of cubic equation. Cardano's formula.
Roots of quartic equation. Cardano's formula.
Rudin's Theorem of Complex Analysis.
Exact values of trigonometric functions of angles (n*pi)/11.
Exact values of trigonometric functions of angles (n*pi)/13.
Exact values of cos(2*k*pi/17).
Exact values of trigonometric functions of angles (n*pi)/17.
Equalities for values of trigonometric functions of angles (n*pi)/17.
Factorization of a polynomial, which defines values of sine function (angles n*pi/17).
Polynomials with roots cos(2*k*pi/n).
Factorization of polynomials with roots cos(2*k*pi/n), where n is Fermat number.
Values of trigonometric functions of angles (n*pi)/257. Part I.
Values of trigonometric functions of angles (n*pi)/257. Part II, Part III, Part IV, Part V, Part VI, Part VII.
Values of trigonometric functions of angles (n*pi)/65537. Part I.
Values of trigonometric functions of angles (n*pi)/65537. Part II, Part III, Part IV and Part V, Part VI, Part VII, Part VIII, Part IX, Part X, Part XI, Part XII, Part XIII, Part XIV.
Positive random walks.
Construction of a regular heptadecagon.
Construction of a regular polygon with 257 sides.


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