Trigonometry Angles

The angles (with integers) for which the trigonometric functions may be expressed in terms of finite root extraction of real numbers are limited to values of which are precisely those which produce constructible polygons. Analytic expressions for trigonometric functions with arguments of this form can be obtained using the Wolfram Language function ToRadicals, e.g., ToRadicals[Sin[Pi/17]], for values of (for , the trigonometric functions 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 (1836).

A necessary and sufficient condition that a regular -gon be constructible is that be a power of 2, where is the totient function (Krížek et al. 2001, p. 34).

Constructible values of for were given by Gauss (Smith 1994), and 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 of for constructible polygons are 1, 1, 1, 1, 2, 1, 2, 2, 2, 4, 4, 8, 4, 4, 4, 8, ...(OEIS A113401), and of are ... (OEIS A113402).

Gardner (1977) and independently Watkins (Conway and Guy 1996, Krížek et al. 2001) noticed that the number of sides for constructible polygons with odd numbers of sides are given by the first 32 rows 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). In other words, every row is a product of distinct Fermat primes, with terms given by binary counting.

A partial table of the analytic values of sine, cosine, and tangent for arguments with small integer is given below. Derivations of these formulas appear in the following entries.

()	(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 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,

(8)

This confirms that for rational, trigonometric functions of are always algebraic numbers. For example, the cases and involve the cubic equation (in and , respectively). The polynomial of which a given expression is a root can be obtained in the Wolfram Language using the syntax RootReduce[ToRadicals[expr]], which produces a Root object.

Letting denoted 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

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).

The minimal polynomial for with an odd prime is given by

(10)

(Beslin and de Angelis 2004).

If and , the algebraic order of is given by

(11)

(Ribenboim 1972, p. 289; Beslin and de Angelis 2004). This 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 with an odd prime, an explicit formula can be given for the minimal polynomial, namely

(14)

where

			(15)
			(16)
			(17)

(Surowski and McCombs 2003; correcting the sign in the definition of ). Watkins and Zeitlin (1993) showed that

(18)

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

(19)

for even.

Beslin and de Angelis (2004) give the simpler form

(20)

where is as defined above.

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 Galois subgroups of 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 example,

			(21)
			(22)
			(23)
			(24)

The first few values of the latter for , 2, ... are therefore 1, 1, 3/4, 1/2, 5/16, 3/16, ... (OEIS A000265 and A084623). Another example is the general case of Morrie's law,

(25)