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

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

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, Part V and Part VI.

Let us define 32 new sets G1, G2, G3,...G16,... G31, G32 in the following way:
The set Gi is a subset of Ei for i=1,2,3...16 and subset of Fi for i=17, 18...32. Moreover sets Gi are 2-elements sets of the form
{n, M(16*n)}, where M is a function defined previously. Exactly:
G1={1, 16}, G17={2, 32},
G2={3, 48}, G18={6, 96},
G3=(5, 80}, G19={10, 97},
G4={7, 112}, G20={14, 33},
G5={9, 113}, G21={18, 31},
G6={11, 81}, G22={22, 95},
G7={13, 49}, G23={26, 98},
G8={15, 17}, G24={30, 34},
G9={19, 47}, G25={38, 94},
G10={21, 79}, G26={42, 99},
G11={23, 111}, G27={46, 35},
G12={25, 114}, G28={50, 29},
G13={27, 82}, G29={54, 93},
G14={37, 79}, G30={74, 101},
G15={43, 83}, G31={86, 91},
G16={45, 51}, G32={90, 102}.

Let us define 32 sets H1, H2,H3,...H16 such that Hi is a difference Ei\Gi for i=1,2...16. For example H11=E11\G11.
Let Hi be the difference Fi\Gi for i=17,18...32.
Let us define Rozmiar: 4888 bajtów

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for i=1, 2...32.
We have Rozmiar: 22963 bajtów

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The same way we show that

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Now we can easily find values of

and

knowing that

for i=1,2,3...16 and

for i=17,18...32.
For example

and

are the roots of the equation

Now we can find any value cos(n*w). For example

so that cos(2*w) and cos(32*w) are the roots of the equation

Formulas for values cos(n*w) cannot be placed on Literka's pages, because they are too large. Later Literka probably will include them as image files.

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

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.

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