Trigonometry Angles

The trigonometry angles considered here are rational multiples of , where is an integer and is a positive integer. The defined values of the trigonometric functions at these angles can be expressed using rational operations and real square roots over the rational numbers iff is a power of 2, where is reduced to lowest terms with positive denominator and is the totient function. For , this is exactly the criterion for a constructible polygon with sides. The values and 2 are degenerate indices rather than ordinary polygon side counts.

Radical expressions can sometimes be obtained using the Wolfram Language function ToRadicals, e.g., ToRadicals[Sin[Pi/17]]. For , the trigonometric functions at auto-evaluate in the Wolfram Language.

Compass and straightedge constructions dating back to Euclid were capable of inscribing regular polygons of 3, 4, 5, 6, 8, 10, 12, 16, 20, 24, 32, 40, 48, 64, ..., sides. However, Gauss showed in 1796 (when he was 19 years old) that a sufficient condition for a regular polygon on sides to be constructible was that be of the form

(1)

where is a nonnegative integer and the are distinct Fermat primes. Here, a Fermat prime is a prime Fermat number, i.e., a prime number of the form

(2)

where is an integer, and the only known primes of this form are 3, 5, 17, 257, and 65537. The first proof of the fact that this condition was also necessary is credited to Wantzel (1837, p. 369).

For , a necessary and sufficient condition that a regular polygon with sides be constructible is that be a power of 2, where is the totient function (Krížek et al. 2001, p. 34).

Values of satisfying the constructibility criterion for were given by Gauss (Smith 1994). Including the degenerate values 1 and 2, the first few are 1, 2, 3, 4, 5, 6, 8, 10, 12, 15, 16, 17, 20, 24, 30, 32, 34, 40, 48, 51, 60, 64, 68, 80, 85, 96, 102, 120, 128, 136, 160, 170, 192, ... (OEIS A003401).

The algebraic degrees over the rational numbers of at the successive constructible values of listed above are 1, 1, 1, 1, 2, 1, 2, 2, 2, 4, 4, 8, 4, 4, 4, 8, ... (OEIS A113401). The corresponding algebraic degrees of are 1, 1, 1, 2, 2, 2, 4, 4, 4, 4, 8, 8, 8, 8, 8, 16, ... (OEIS A113402).

Gardner (1977) and independently Watkins (Conway and Guy 1996, Krížek et al. 2001) noticed that the odd constructible values obtained from the five known Fermat primes are given by rows 0 through 31 of the Sierpiński sieve interpreted as binary numbers, giving 1, 3, 5, 15, 17, 51, 85, 255, ... (OEIS A004729; Conway and Guy 1996, p. 140). These 32 rows are the products of subsets of the five known Fermat primes, ordered by binary counting. Row 0 is the empty product 1, which is not a polygon side count. An additional Fermat prime, if one exists, would give further odd constructible side counts.

A partial table of the analytic values of sine, cosine, and tangent at common angles, including for several small positive integers , is given below. The symbol at 90 degrees denotes a pole, not a defined real value of the tangent function.

()	(rad)
0.0	0	0	1	0
15.0
18.0
22.5
30.0
36.0
45.0				1
60.0
90.0		1	0
180.0		0		0

There is a nice mnemonic for remembering sines of common angles,

			(3)
			(4)
			(5)
			(6)
			(7)

In general, any defined trigonometric function can be expressed in radicals for arguments of the form , where is a rational number, by writing the trigonometric functions in exponential form and the exponentials as roots of . For example, using the principal value of complex exponentiation,

(8)

This confirms that for rational , defined trigonometric functions of are always algebraic numbers. For example, the cases and involve the cubic equation (in and , respectively). An algebraic number can be reduced to a canonical form using RootReduce[expr], often producing a Root object. A primitive integer polynomial of which it is a root can be obtained in the Wolfram Language using MinimalPolynomial[expr, x].

Letting denote the th root of the polynomial in the ordering of the Wolfram Language's Root object, the first few analytic values of are summarized in the following table.

1	0
2	1
3
4
5
6
7
8
9
10

Let denote the polynomial degree of the minimal polynomial of an algebraic number over the rational numbers. For a positive integer , the algebraic order of is given analytically by

(9)

where is the totient function. For , 2, ..., this gives the sequence 1, 1, 2, 2, 4, 1, 6, 4, 6, 2, 10, 4, ... (OEIS A055035).

A primitive integer multiple of the minimal polynomial for with an odd prime is given by

(10)

(Beslin and de Angelis 2004).

For a positive integer , write with a nonnegative integer and an odd number. The algebraic order of is given by

(11)

(Ribenboim 1972, p. 289; Beslin and de Angelis 2004). The excluded case has degree 1. Including it gives the sequence 1, 1, 2, 1, 4, 2, 6, 2, 6, 4, 10, 1, ... (OEIS A093819).

The first few analytic values of are summarized in the following table.

1
2	0
3
4
5
6
7
8
9
10

The algebraic order of is given analytically by

(12)

where is the totient function. For , 2, ..., this gives the sequence 1, 1, 1, 2, 2, 2, 3, 4, 3, 4, 5, 4, 6, ... (OEIS A055034; Lehmer 1933, Watkins and Zeitlin 1993, Surowski and McCombs 2003).

The algebraic order of is given analytically by

(13)

(Ribenboim 1972, p. 289; Beslin and de Angelis 2004) giving the sequence 1, 1, 1, 1, 2, 1, 3, 2, 3, 2, 5, ... (OEIS A023022).

For a positive integer , let be the monic minimal polynomial of over the rational numbers. In particular, and . For an odd prime , an explicit formula is

(14)

where and, for the indices appearing in the sum,

			(15)
			(16)

(Surowski and McCombs 2003). Watkins and Zeitlin (1993) showed that

(17)

for odd, where is a Chebyshev polynomial of the first kind, and

(18)

for even. The products include all positive divisors of .

Beslin and de Angelis (2004) give a primitive integer multiple of in the simpler form

(19)

where is as defined above. Its monic normalization is .

As already noted, a special type of expansion in terms of radicals with real arguments can be obtained if is a power of two times a product of distinct Fermat primes. For other values of , the situation becomes more complicated. It is now no longer possible to express trigonometric functions in a form that they are expressed as real radicals, but a certain minimal representation still exists. The simplest nontrivial example is for . The exact meaning of "minimal" is rather technical and is related to the subgroups of the Galois groups associated with certain cyclotomic polynomials (Weber 1996). As it turns out, for prime, the expansions are especially interesting and difficult, and higher order Galois group calculations are both difficult and time-consuming. For example, is a very difficult case and takes a long time to calculate. Some larger primes are easier again but the complexity grows with the size of the prime on average.

While individual trigonometric functions may require complicated representations at certain angles, there are general formulas for the products of these functions. For a positive integer , for example,

			(20)
			(21)
			(22)
			(23)

Here denotes the floor function. The first few values of the square of the latter product for , 2, ... are therefore 1, 1, 3/4, 1/2, 5/16, 3/16, ... (OEIS A000265 and A084623 for the reduced numerators and denominators, respectively). For a nonnegative integer and real with , the general case of Morrie's law is

(24)