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Factorization of a Polynomial, which Defines Values of Sine Function
(Angles n*pi/17).


by 'Literka'.


In the following text we'll be using the notations pi=3.14159... and w=pi/17.


Simple computations and the formula of DeMoivre shows that the squares of exact values of sin(n*w) (where n=1,2..8) are the roots of the polynomial

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Let us denote this polynomial by W(x). It is hard to see how to find roots of W(x). If we can write W as W(x)=P(x)*Q(x), where P(x) and Q(x) are forth degree polynomials, then we can find roots of P(x) and Q(x) using the formula of Cardano. Of course, roots of P(x) and Q(x) are also roots of W(x).
To find P(x) and Q(x) let us notice that W(x) is equal to

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This shows that we can take P(x) equal to

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and we can take Q(x) equal to

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Now we can find easily roots of W(x) using Cardano's formula. Let us skip it, since these values are shown on the page Values of trigonometric functions.
We'll try to figure out some consequences of the derived factorization. We need to know that the roots of polynomial P(x) are squares of sin(w), sin(2w), sin(4w), sin(8w) and that the roots of Q(x) are squares of sin(3w), sin(5w), sin(6w), sin(7w). The formula of Vieta applied to P(x) and Q(x), after taking square roots of each side, gives us:
The value of
sin(w)*sin(2w)*sin(4w)*sin(8w)

is equal to
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The value of
sin(3w)*sin(5w)*sin(6w)*sin(7w)

is equal to
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There are other consequences of formula of Vieta, but some of them follow from the equalities of the page Equalities of trigonometric functions.



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

See other pages of 'Literka' from Mathematical Countryside:
 Monotonic subsequences.
A remarkable monotonic property of the gamma function .
Weight centers of simple geometrical figures.
An elementary problem can be unsolvable.
Construction of a regular pentagon.
Construction of a regular heptadecagon.
Construction of a regular polygon with 257 sides.
Roots of cubic equation. Cardano's formula.
Rudin's Theorem of Complex Analysis.
Weierstrass Approximation Theorem.
Exact values of trigonometric functions of angles (n*pi)/17.
Equalities for values of trigonometric functions of angles (n*pi)/17.
Polynomials with roots cos(2*k*pi/n).
Factorization of polynomials with roots cos(2*k*pi/n), where n is Fermat number.
Values of trigonometric functions of angles (n*pi)/257. Part I.
Values of trigonometric functions of angles (n*pi)/257. Part II, Part III, Part IV, Part V, Part VI, Part VII.
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.

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