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

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

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 4 previous pages of Literka Part I , Part II , Part III and Part IV.
To see further development of the current page click Values of trigonometric functions. Part VI and Part VII.

Let us introduce 16 new numbers

to be the summations of terms cos(i*w), where i is and element of Ai (Sets A1, A2,...A16 were defined previously). For example
Rozmiar: 2726 bajtów

The definitions of

(we already know the values of these numbers from the previous parts of this page) show that these numbers are linear combinations of

.
For example

Hence, regarding

as unknown variables, we have 15 linear equations with 16 unknowns. We add one more equation (as the first one): Rozmiar: 5865 bajtów

We obtain a system of 16 equations. The matrix of this system is:

Rozmiar: 24663 bajtów

The determinant of this matrix is 34816, which is a product of a power of 2 and the number 17. This is little surprising, since 17 is also a Fermat number just preceding the number 257. Because of the value of determinant, the number 17 will occur in the next formulas.
The inverse matrix to the above matrix is:
Rozmiar: 48933 bajtów

Using this matrix it is easy to write formulas for

. For example, let us write a formula for i=10:

Rozmiar: 77648 bajtów

with the notations of a page of Literka Part IV.

See the next part of the current page Values of trigonometric functions. Part VI and Part VII

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

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.

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'.