REDUCTION FORMULAS OF THE COSINE OF INTEGER FRACTIONS OF π

Transcription

1 REDUCTION FORMULAS OF THE COSINE OF INTEGER FRACTIONS OF π RICHARD J. MATHAR Abstract. The power of some cosines of integer fractions π/n of the half circle allow a reduction to lower powers of the same angle. These are tabulated in the format n/2 X a (n) i cos i π = 0; (n = 2, 3, 4,...). n i=0 Related expansions of Chebyshev Polynomials T n(x) and factorizations of T n(x) + 1 are also given. 1. Description 1.1. Chebyshev Coefficients. Each first formula in the list given in Section 2 is an expansion [7, ] of (1) n/2 cos(nϕ) = 2 n 1 cos n ϕ + n ( 1) k 1 ( ) n k 1 2 n 2k 1 cos n 2k ϕ; (n > 0), k k 1 k=1 providing an extension of the Tables [7, 1.335] which cover the cases up to n = 7. With (2) Re inϕ = cos(nϕ) = T n (x), this is also an extension of the Table [1, 22.3] of the coefficients of the Chebyshev Polynomials [1, ][2, (7)], and of sequence A of the OEIS [10]. The shortcut x cos ϕ is used throughout Cosine Reductions. Each second formula for a n is a Gröbner-base reduction [8] of the equation cos(nϕ) = 1 for ϕ = π/n, i.e., obtained through factorization of the polynomial T n (x) + 1 trivially derived from each first formula. Example: For n = 9, Eq. (11) the factorization is T 9 (x) + 1 = 256x 9 576x x 5 120x 3 + 9x + 1 = (x + 1)(2x 1) 2 (8x 3 6x 1) 2. The existence of a non-trivial factorization is demonstrated via [1, ] and [1, ]: (3) T n (x)+1 = T 2m (x) + 1 = 2Tm(x) 2 ; (n 2m, even); T 2m+1 (x) + 1 = (x + 1)[ 2m 1 i=0 2( 1) i (i + 1)T 2m i (x) + (2m + 1)] = (x + 1)[1 2(1 x) m 1 k=0 U 2k+1(x)], (n 2m + 1, odd). Date: October 29, Key words and phrases. trigonometric functions. 1

2 2 RICHARD J. MATHAR If n is not a prime number, the second line tabulated comprises the formula which appears for a multiple earlier in the table implicitly. Example: Eq. (14) for n = 14 is an implicit equation for cos(π/14) which is reduced to Eq. (10) by use of the duplication formula [7, 1.323] cos 2 π 14 = 1 2 [1 + cos π 7 ] and its powers cos 4 π 14 = 1 4 [1 + cos π 7 ]2 and cos 6 π 14 = 1 8 [1 + cos π 7 ]3. Construction of the results would also work for composite n in the other direction. Higher cosine powers than the leading power of the second equations follow by taking the remainder of a polynomial division modulo the one actually shown. Example: Eq. (13) has a leading power 4 in cos π 12, so polynomials of degree 4 and higher can be reduced to polynomials of degree 3 or less: (4) cos 5 π 12 = 1 16 cos π 12 [16 cos 4 π cos2 π ] +cos 3 π cos π 12 = cos3 π cos π 12 ; (5) cos 6 π [ 1 12 = π ] [ 16 cos2 16 cos 4 π π ] cos π 16 cos = 15 π 16 cos Representations by Irrational Numbers. The table is also a quick guide to construct cosines in terms of square and cubic roots of rational numbers: if the second equation is quadratic, bi-quadratic etc. in cos(π/n), it can be solved for cos(π/n) with the standard formulas for this type of polynomial roots [9]. Example: Eq. (12) is bi-quadratic in cos(π/10), with a first root (6) cos 4 π cos2 π = 0 cos2 π 10 = and therefore a second root (7) cos π =. 8 ( 5 8 ) = , Bootstrapping from the expressions for cos π 2, cos π 3, cos π 5 and cos π 15 given below, and with [1, ] (8) cos z 1 + cos z 2 = ±, 2 one finds algebraic representations for all cosines of the form cos π, cos, 2 k 3 2 k π π cos, and cos, k = 0, 1, 2,... [3, 6]. Explicit values for n = 5, 8, 12, k k have been given by Vidūnas [11, 12]. We adopt his notation: (9) φ 1 + 5; φ 1 5; ψ ; ψ The values of ψ and ψ are entries A and A of the OEIS [10]. All but the first of the decimal digits of φ and φ are those of entry A Other ways to combine these square roots are known [4, 5]. π n = 2 2. Table Re 2iϕ = 2x 2 1; cos π 2 = 0.

3 REDUCTION FORMULAS OF THE COSINE OF INTEGER FRACTIONS OF π 3 n = 3 n = 4 n = 5 Re 3iϕ = 4x 3 3x; 2 cos π 3 1 = 0. cos π 3 = 1 2. Re 4iϕ = 8x 4 8x 2 + 1; 2 cos 2 π 4 1 = 0. is entry A in [10]. cos π 4 = 1 2 Re 5iϕ = 16x 5 20x 3 + 5x; 4 cos 2 π 5 2 cos π 5 1 = 0. n = 6 n = 7 is entry A in [10]. cos π 5 = φ 4 Re 6iϕ = 32x 6 48x x 2 1; 4 cos 2 π 6 3 = 0. is entry A in [10]. cos π 6 = 3 2 (10) Re 7iϕ = 64x 7 112x x 3 7x; 8 cos 3 π 7 4 cos2 π 7 4 cos π = 0. n = 8 n = 9 cos π 7 is entry A in [10]. Re 8iϕ = 128x 8 256x x 4 32x 2 + 1; 8 cos 4 π 8 8 cos2 π = 0. cos π 8 = is entry A in [10]. (11) Re 9iϕ = 256x 9 576x x 5 120x 3 + 9x; 8 cos 3 π 9 6 cos π 9 1 = 0. n = 10 (12) cos π 9 is entry A in [10]. Re 10iϕ = 512x x x 6 400x x 2 1; 16 cos 4 π π cos = 0. n = 11 cos π 10 = is entry A in [10]. Re 11iϕ = 1024x x x x x 3 11x; 32 cos 5 π cos4 π cos3 π cos2 π cos π 11 1 = 0.

4 4 RICHARD J. MATHAR n = 12 Re 12iϕ = 2048x x x x x 4 72x 2 + 1; (13) n = cos 4 π cos2 π = 0. is entry A in [10]. cos π 12 = Re 13iϕ = 4096x x x x x 5 364x x; 64 cos 6 π π cos π cos π cos π cos cos π 13 1 = 0. n = 14 Re 14iϕ = 8192x x x x x x x 2 1; (14) 64 cos 6 π cos4 π cos2 π 14 7 = 0. n = 15 Re 15iϕ = 16384x x x x x x x 3 15x; 16 cos 4 π cos3 π cos2 π 15 8 cos π = 0. n = 16 is entry A in [10]. cos π 15 = φ 3ψ Re 16iϕ = 32768x x x x x x x 4 128x 2 + 1; 128 cos 8 π cos6 π cos4 π cos2 π = 0. cos π 1 16 = n = 17 Re 17iϕ = 65536x x x x x x x 5 816x x; 256 cos 8 π π cos π cos π cos π cos π cos cos 2 π cos π = 0. n = 18 Re 18iϕ = x x x x x x x x x 2 1; 64 cos 6 π π cos π cos = 0. cos π 18 is entry A in [10].

5 REDUCTION FORMULAS OF THE COSINE OF INTEGER FRACTIONS OF π 5 n = 19 Re 19iϕ = x x x x x x x x x 3 19x; 512 cos 9 π π cos π cos π cos π cos π cos cos 3 π π cos cos π 19 1 = 0. n = 20 Re 20iϕ = x x x x x x x x x 4 200x 2 + 1; 256 cos 8 π π cos π cos π cos = 0. cos π 20 = φ ψ n = 21 is entry A in [10]. Re 21iϕ = x x x x x x x x x x x; 64 cos 6 π π cos π cos π cos π cos cos π = 0. n = 22 Re 22iϕ = x x x x x x x x x x x 2 1; 1024 cos 10 π π cos π cos π cos π cos2 11 = n = 23 Re 23iϕ = x x x x x x x x x x x 3 23x; 2048 cos 11 π π cos π cos π cos π cos π cos cos 5 π π cos π cos π cos cos π = 0. n = 24 Re 24iϕ = x x x x x x x x x x x 4 288x 2 + 1; 256 cos 8 π π cos π cos π cos = 0. is entry A in [10]. cos π 24 = /4

6 6 RICHARD J. MATHAR n = 25 Re 25iϕ = x x x x x x x x x x x x x; 1024 cos 10 π π cos π cos π cos π cos π cos cos 2 π cos π 25 1 = 0. n = 26 Re 26iϕ = x x x x x x x x x x x x x 2 1; 4096 cos 12 π π cos π cos π cos π cos π cos = n = 27 Re 27iϕ = x x x x x x x x x x x x x 3 27x; 512 cos 9 π π cos π cos π cos cos π 27 1 = 0. n = 28 Re 28iϕ = x x x x x x x x x x x x x 4 392x 2 + 1; 4096 cos 12 π π cos π cos π cos π cos π cos = 0. n = 29 Re 29iϕ = x x x x x x x x x x x x x x x; cos 14 π π cos π cos π cos π cos π cos cos 8 π π cos π cos π cos π cos π cos cos 2 π cos π 29 1 = 0. n = 30 Re 30iϕ = x x x x x x x x x x x x x x x 2 1; 256 cos 8 π π cos π cos π cos = 0. cos π 30 is entry A in [10].

7 REDUCTION FORMULAS OF THE COSINE OF INTEGER FRACTIONS OF π 7 n = 31 Re 31iϕ = x x x x x x x x x x x x x x x 3 31x; cos 15 π π cos π cos π cos π cos cos 10 π π cos π cos π cos π cos cos 5 π π cos π cos π cos cos π = 0. n = 32 Re 32iϕ = x x x x x x x x x x x x x x x 4 512x 2 + 1; cos 16 π π cos π cos π cos π cos cos 6 π π cos π cos = 0. n = 33 Re 33iϕ = x x x x x x x x x x x x x x x x x; 1024 cos 10 π π cos π cos π cos π cos cos 5 π π cos π cos π cos cos π = 0. n = 34 Re 34iϕ = x x x x x x x x x x x x x x x x x 2 1; cos 16 π π cos π cos π cos π cos cos 6 π π cos π cos = References 1. Milton Abramowitz and Irene A. Stegun (eds.), Handbook of mathematical functions, 9th ed., Dover Publications, New York, MR (29 #4914) 2. Richard Barakat, Evaluation of the incomplete gamma function of imaginary argument by Chebyshev polynomials, Math. Comp. 15 (1961), no. 73, 7 11, E: The factors (iq) ν in (5), (9), (13) and (14) should read (ix) ν. The values for the Newmann s factor after (6) need to be swapped. MR MR (23 #B1103)

8 8 RICHARD J. MATHAR 3. Paul Bracken and Jiri Čižek, Evaluation of quantum mechanical perturbative sums in terms of quadratic surds and their use in approximation of ζ(3)/π 3, Int. J. Quant. Chem. 90 (2002), no. 1, John H. Conway, Charles Radin, and Lorenzo Sadun, On angles whose squared trigonometric functions are rational, arxiv:math-ph/ (1998). 5., On angles whose squared trigonometric functions are rational, Disc. Comput. Geom. (1999), no. 22, Kurt Girstmair, Some linear relations between values of trigonometric functions at kπ/n, Acta Arithm. 81 (1997), MR (98h:11133) 7. I. Gradstein and I. Ryshik, Summen-, Produkt- und Integraltafeln, 1st ed., Harri Deutsch, Thun, MR (83i:00012) 8. S. Gurak, On the minimal polynomial of gauss periods for prime powers, Math. Comp. 75 (2006), no. 256, MR (2007c:11088) 9. L. D. Servi, Nested square roots of 2, Am. Math. Monthly 110 (2003), no. 4, MR Neil J. A. Sloane, The On-Line Encyclopedia Of Integer Sequences, Notices Am. Math. Soc. 50 (2003), no. 8, , njas/sequences/. MR (2004f:11151) 11. Raimundas Vidūnas, Expressions for values of the gamma function, arxiv:math.ca/ (2004). 12., Expressions for values of the gamma function, Kyushu J. Math 59 (2005), no. 2, MR (2007b:33005) address: Leiden Observatory, P.O. Box 9513, 2300 RA Leiden, The Netherlands