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

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

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 from the page of Literka Values of trigonometric functions. Part I. On this page sets A, B, C were defined and the sums of cosines corresponding to these sets were computed. On the current page we'll do the same, but for different sets. This will approach us to achieve the final result.
See Values of trigonometric functions. Part III , Part IV, Part V, Part VI and Part VII for further development of the current page.

Let us define set B1 to be the union of the sets (defined before on the quoted page) A3, A5, A7, A10, A12, A13, A14, A15.
Let us define set C1 to be all other elements of the set A (i.e. it a difference of the sets A and B1).
Finally let Rozmiar: 4146 bajtów

Using previous notations and calculations Rozmiar: 2211 bajtów

Moreover

Comments for the proof of this equality are the same as for s and t on the page of Literka Values of trigonometric functions. Part I. Using the computed values there we obtain Rozmiar: 7865 bajtów

Hence, t1 and s1 are the roots of the quadratic equation: Rozmiar: 2507 bajtów

Solving it and verifying values we receive: Rozmiar: 7181 bajtów

Similarly let us define set B2 to be the union of the sets A2, A4, A5, A7, A9, A10, A12, A16.
Let C2 be the difference of sets A and B2 and Rozmiar: 4157 bajtów

By the same arguments as before

Rozmiar: 1784 bajtów

and

The same way as before we derive formulas of this page: Rozmiar: 6714 bajtów

Finally, let us write all derived formulas:

Rozmiar: 20753 bajtów

It is still far away from our final goals, but it is getting closer.

See next parts of this page
Part III, Part IV, Part V, Part VI, Part VII.

Have you ever seen formula 9000 feet long?
Another way to find cos(pi/257).
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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.
Values of trigonometric functions of angles (n*pi)/65537. Part II, Part III, Part IV Part V, Part VI, Part VII, Part VIII, Part IX, Part X, Part XI, Part XII, Part XIII, Part XIV.
Polynomials with roots cos(2*k*pi/n).
Factorization of polynomials with roots cos(2*k*pi/n), where n is Fermat number.
An elementary problem can be unsolvable.
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.

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