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Powers of Cosine and Sine

Leon D. Fairbanks

TL;DR

This work develops a comprehensive algebraic framework for expressing sums of powers of sine and cosine at dyadic angles in terms of the rational basis generated by adjoining these trigonometric values to the rationals. It constructs minimal polynomials for cosine values of the form cos((2i-1)π/2^n) with degree 2^{n-1} and shows how odd and even powers can be represented in the basis via explicit 2^{n-2}×2^{n-2} matrices, aided by Chebyshev-type polynomials p_i that express sine values in terms of cosine. The approach yields detailed reciprocal-power identities, highlights the permutation/automorphism structure of the associated field extensions, and culminates in a new zeta-function representation in terms of dyadic-angle sums, connecting to Apery’s constant and Bernoulli numbers. An accompanying appendix provides Mathematica code to construct the required matrices and verify the algebraic relations. Overall, the paper blends field theory, combinatorics, and computational methods to derive explicit, basis-based expressions for trigonometric sums and their connections to zeta values.

Abstract

This paper presents expressions for sums of powers of sine and cosine in terms of the basis for the field extension obtained by adjoining the sine or cosine to the field of rational numbers.

Powers of Cosine and Sine

TL;DR

This work develops a comprehensive algebraic framework for expressing sums of powers of sine and cosine at dyadic angles in terms of the rational basis generated by adjoining these trigonometric values to the rationals. It constructs minimal polynomials for cosine values of the form cos((2i-1)π/2^n) with degree 2^{n-1} and shows how odd and even powers can be represented in the basis via explicit 2^{n-2}×2^{n-2} matrices, aided by Chebyshev-type polynomials p_i that express sine values in terms of cosine. The approach yields detailed reciprocal-power identities, highlights the permutation/automorphism structure of the associated field extensions, and culminates in a new zeta-function representation in terms of dyadic-angle sums, connecting to Apery’s constant and Bernoulli numbers. An accompanying appendix provides Mathematica code to construct the required matrices and verify the algebraic relations. Overall, the paper blends field theory, combinatorics, and computational methods to derive explicit, basis-based expressions for trigonometric sums and their connections to zeta values.

Abstract

This paper presents expressions for sums of powers of sine and cosine in terms of the basis for the field extension obtained by adjoining the sine or cosine to the field of rational numbers.
Paper Structure (8 sections, 64 theorems, 183 equations)

This paper contains 8 sections, 64 theorems, 183 equations.

Table of Contents

  1. Introduction
  2. Odd Powers Of Cosine
  3. Even Powers Of Cosine
  4. Negative Odd Powers
  5. Some Trig Results
  6. An Interesting Formula For The Riemann Zeta Function
  7. Appendix
  8. Bibliography

Key Result

Lemma 1.2

Assume $r,k\in \mathbb{Z}$, $1\le k\le r$, then

Theorems & Definitions (113)

  • Definition 1.1
  • Lemma 1.2
  • proof
  • Corollary 1.3
  • Proposition 1.4
  • proof
  • Lemma 2.1
  • proof
  • Lemma 2.2
  • proof
  • ...and 103 more