Values of Trigonometric Functions of Angles (n*pi)/257. Part VI.

Values of Trigonometric
Functions of Angles (n*pi)/257.
Part VI.

by 'Literka'.

In the following text we'll be using the notations pi=3.14159... and w=(2*pi)/257.

We'll be using all notations of 5 previous pages of Literka Part I , Part II , Part III , Part IV and Part V.
To see further development of the current page click Values of trigonometric functions. Part VII.

Let us define 16 new sets E1, E2, E3,...E13, E14, E15, E16 in the following way:
The set Ei is a subset of Ai for i=1,2,3...16 (for example E10 is a subset of A10).
Each Ei is of the form {x, M(4*x), M(16*x), M(64*x)}, where M is a function defined on the first page. Exactly:

E1={1, 4, 16, 64}
E2={3, 12, 48, 65}
E3={5, 20, 80, 63}
E4={7, 28, 112, 66}
E5={9, 36, 113, 62}
E6={11, 44, 81, 67}
E7={13, 52, 49, 61}
E8={15, 60, 17, 68}
E9={19, 76, 47, 69}
E10={21, 84, 79, 59}
E11={23, 92, 111, 70}
E12={25, 100, 114, 58}
E13={27, 108, 82, 71}
E14={37, 109, 78,55}
E15={43, 85, 83, 75}
E16={45, 77, 51, 53}

Let Fi (i=1,2..16) be differences Ai\Ei (for example F11=A11\E11).
Finally, let us define Rozmiar: 5099 bajtów

It follows that

We can verify (this time easily since we have only 32 terms) that
Rozmiar: 16660 bajtów

We see that x1 and y1 are the roots of the equation: Rozmiar: 4107 bajtów

Using a matrix of the page Part V we compute:

Rozmiar: 20533 bajtów

On the right hand sides of these equalities there are only known numbers. Hence, we compute the roots of the above quadratic equation, hence x1 and y1. Literka will not show formulas for these numbers, because they are not very readable. Literka will attach them in a later time.
Similarly we can obtain 15 other equations: Rozmiar: 44370 bajtów

Roots of the first one of these equations are x2 and y2, roots of the second equation are x3 and y3 ...etc.
From these we can deduce formulas for x2, x3... x16, and y2, y3,...y16. As before, we'll not present these formulas for this time.
Let us notice that we almost achieved our goals. Still there are two steps left to do. There are on the page

See a final page of this theory Part VII.

See the previous parts of this page
Part I , Part II , Part III, Part IV, Part V.

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

See other pages of 'Literka' from Mathematical Countryside:
A Remarkable Monotonic Property of the Gamma Function .
Monotonic subsequences.
Weight centers of simple geometrical figures.
Roots of cubic equation. Cardano's formula.
Rudin's Theorem of Complex Analysis.
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).
Values of trigonometric functions of angles (n*pi)/65537. Part I, Part II.
Values of trigonometric functions of angles (n*pi)/65537. Part III and Part IV, Part V, Part VI, Part VII Part VIII, Part IX, Part X, Part XI, Part XII, Part XIII, Part XIV.
An elementary problem can be unsolvable.
Polynomials with roots cos(2*k*pi/n).
Factorization of polynomials with roots cos(2*k*pi/n), where n is Fermat number.
Weierstrass Approximation Theorem. Bernstein's Polynomials.
Construction of a regular pentagon.
Construction of a regular heptadecagon.
Construction of a regular polygon with 257 sides.

Software that you really need.

Return to the list of pages of 'Literka' about polytopes.
Return to the main geometrical page of 'Literka'.
Return to the main page of 'Literka'.