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


by 'Literka'.


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


The number 257 is the next Fermat number after 17. It is known that if n is prime and a Fermat number (i.e. prime number of the form

then a regular polygon with n sides can be constructed with a ruler and compass. This was proved by Gauss. For that reason we shall try to derive formulas for trigonometric values of angles n*pi/257. There is a page of Chris Becker showing a formula for sin(pi/257). However there is nothing about how to derive this formula.
The current page is only an introduction for deriving interesting us formulas. We derive some equalities, but the main purpose is still far away. Our considerations are based on the page of Satoshi Hoshino. Unfortunately this page is written in Japanese language and we can only hope that it will be translated into English.
Notice: Recently this page was translated into English.
More formulas for values of trigonometric functions (angles pi/257) can be found on the next parts of this page Part II, Part III, Part IV, Part V, Part VI, Part VII.

Let A be a set of all integers from 1 to 128, A={1,2,3,4...128}. Let M(x) be a function defined on positive integers such that
  1. M(x)=x for 129>x
  2. M(x)=257-x for 257>x>128
  3. M(x)=M(y) for x>=257, where y= x mod 257 (the rest of x of division by 257)

It follows that M(x+257)=M(x)=M(257-x) (We can also define M for negative integers)

For a positive integer x let us take subsets of A of the form
{ M(x), M(2*x), M(4*x), M(8*x), M(16*x), M(32*x), M(64*x), M(128*x) }

This will split the set A into 16 sets A1, A2,.. A16:
A1= {1, 2, 4, 8, 16, 32, 64, 128 }
A2= { 3, 6, 12, 24, 48, 96, 65, 127 }
A3= { 5, 10, 20 ,40, 80, 97, 63, 126 }
A4= { 7, 14, 28, 56, 112, 33, 66, 125 }
A5= { 9, 18, 36, 72, 113, 31,62,124 }
A6= { 11, 22, 44, 88, 81, 95, 67, 123 }
A7= { 13, 26, 52, 104, 49, 98, 61, 122 }
A8= { 15, 30, 60, 120, 17, 34, 68, 121 }
A9= { 19, 38, 76, 105, 47, 94, 69, 119 }
A10= { 21, 42, 84, 89, 79, 99, 59, 118 }
A11= { 23, 46, 92, 73, 111, 35, 70, 117 }
A12= { 25, 50, 100, 57, 114, 29, 58, 116 }
A13= { 27, 54, 108, 41, 82, 93, 71, 115 }
A14= { 37, 74, 109, 39, 78, 101, 55, 110 }
A15= { 43, 86, 85, 87, 83, 91, 75, 107 }
A16= { 45, 90, 77, 103, 51, 102, 53, 106 }

Let us define the set B as the union of sets A2, A3, A4, A9, A13, A14, A15, A16. Let us define the set C as the set of all other elements from A, i.e. C=A\B (difference of A and B). Hence, A is a union of disjoint sets B and C, each of them having 64 elements. We have an equality (which is the main idea of this page)
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This equality can be deduced just by checking. The left hand side is the sum of expressions of the form
cos(aw)*cos(bw)
where a and b are integers. Instead of this values we can substitute equal values
(cos((a+b)*w)+cos((a-b)*w))/2.
Then we can apply the fact that cos(n*w)=cos(M(n)*w). After reducing similar terms, we receive the above equality. The problem is that we need to do few thousands substitutions. That is why it is better to find another method or rely on Literka, since Literka checked it and it was correct.
For every positive integer n and real v we have
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Proof of this can be found on the page of Satoshi Hoshino. It is a basic equality in the theory of trigonometric series. This equality can also be proved by induction.
Applying this equality for v=w we receive
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Let us introduce notations:
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Knowing that union of B and C is A and combining all previous equalities we can write
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Hence, t and s are the roots of the equation
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having roots
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It can be verified that t is negative, hence
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We received some non-obvious equalities, but it is only a good starting point.

See the next parts of this page
Part II, 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).
Software of Literka, "Mathematical Reader" Download free.



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.
Rudin's Theorem of Complex Analysis.
Roots of cubic equation. Cardano's formula.
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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