Trigonometry Angles

The angles mpi/n (with m,n integers) for which the trigonometric functions may be expressed in terms of finite root extraction of real numbers are limited to values of m 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 n>=7 (for n<=6, 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 n sides to be constructible was that n be of the form


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


where n>=0 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 n-gon be constructible is that phi(n) be a power of 2, where phi(n) is the totient function (Krížek et al. 2001, p. 34).

Constructible values of n for n<300 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 cos(2pi/n) for constructible polygons are 1, 1, 1, 1, 2, 1, 2, 2, 2, 4, 4, 8, 4, 4, 4, 8, ...(OEIS A113401), and of cos(pi/n) 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 pi/n with small integer n is given below. Derivations of these formulas appear in the following entries.

x ( degrees)x (rad)sinxcosxtanx

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

sin(0 degrees)=1/2sqrt(0)
sin(30 degrees)=1/2sqrt(1)
sin(45 degrees)=1/2sqrt(2)
sin(60 degrees)=1/2sqrt(3)
sin(90 degrees)=1/2sqrt(4).

In general, any trigonometric function can be expressed in radicals for arguments of the form rpi, where r=m/n is a rational number, by writing the trigonometric functions in exponential form and the exponentials as roots of -1. For example,


This confirms that for r rational, trigonometric functions of rpi are always algebraic numbers. For example, the cases n=7 and n=9 involve the cubic equation (in sin^2(pi/7) and sin^2(pi/9), 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 (P(x))_n denoted the nth root of the polynomial P(x) in the ordering of the Wolfram Language's Root object, the first few analytic values of sin(pi/n) are summarized in the following table.


The algebraic order of sin(pi/n) is given analytically by

 ord(sin(pi/n))={1   if n=2; phi(n)   if n=0,1,3 (mod 4); 1/2phi(n)   otherwise,

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

The minimal polynomial for sin(2pi/p) with p an odd prime is given by

 S_p(x)=sum_(k=0)^((p-1)/2)(-1)^k(p; 2k+1)(1-x^2)^((p-1)/2-k)x^(2k)

(Beslin and de Angelis 2004).

If n!=4 and n=2^rm, the algebraic order of sin(2pi/n) is given by

 ord(sin((2pi)/n))={phi(n)   if r=0, 1; 1/4phi(n)   if r=2; 1/2phi(n)   if r>=3

(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 cos(pi/n) are summarized in the following table.


The algebraic order of cos(pi/n) is given analytically by

 ord(cos(pi/n))={1   if n=1; phi(n)   if n=0 (mod 2); 1/2phi(n)   if n=1 (mod 2),

where phi is the totient function. For n=1, 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 cos(2pi/n) is given analytically by

 ord(cos((2pi)/n))={1   if n=1 or 2; 1/2phi(n)   if n>2

(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 cos(2pi/p) with p an odd prime, an explicit formula can be given for the minimal polynomial, namely



sigma_(2k)=(-1)^k(s-k; k)
sigma_(2k-1)=(-1)^k(s-k; k-1)

(Surowski and McCombs 2003; correcting the sign in the definition of sigma_(2k-1)). Watkins and Zeitlin (1993) showed that


for n=2s+1 odd, where T_n(x) is a Chebyshev polynomial of the first kind, and


for n=2s even.

Beslin and de Angelis (2004) give the simpler form


where S_p(x) is as defined above.

As already noted, a special type of expansion in terms of radicals with real arguments can be obtained if n is a power of two times a product of distinct Fermat primes. For other values of n, 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 n=7. 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 n prime, the expansions are especially interesting and difficult, and higher order Galois group calculations are both difficult and time-consuming. For example, n=23 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,

product_(k=1)^(n)cos((kpi)/n)={0 if n=0 (mod 2); ((-1)^((n+1)/2))/(2^(n-1)) if n=1 (mod 2)
product_(k=1)^(|_n/2_|)cos((kpi)/n)={0 if n=0 (mod 2); 1/(2^((n-1)/2)) if n=1 (mod 2)

The first few values of the latter for n=1, 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,


See also

257-gon, 65537-gon, Constructible Polygon, Cyclotomy, Fermat Prime, Heptadecagon, Morrie's Law, Pentagon, Sierpiński Sieve, Trigonometry Angles--0, Trigonometry Angles Pi, Trigonometry Angles--Pi/2, Trigonometry Angles--Pi/3, Trigonometry Angles--Pi/4, Trigonometry Angles--Pi/5, Trigonometry Angles--Pi/6, Trigonometry Angles--Pi/7, Trigonometry Angles--Pi/8, Trigonometry Angles--Pi/9, Trigonometry Angles--Pi/10, Trigonometry Angles--Pi/11, Trigonometry Angles--Pi/12, Trigonometry Angles--Pi/13, Trigonometry Angles--Pi/15, Trigonometry Angles--Pi/16, Trigonometry Angles--Pi/17, Trigonometry Angles--Pi/18, Trigonometry Angles--Pi/20, Trigonometry Angles--Pi/23, Trigonometry Angles--Pi/24, Trigonometry Angles--Pi/30, Trigonometry Angles--Pi/32, Unit Circle

Explore with Wolfram|Alpha


Bachmann, P. Die Lehre von der Kreistheilung. Leipzig, Germany: Teubner, 1872.Bauer, F. L. "Sternpolygone und Hyperwürfel." In Miscellanea Mathematica (Ed. P. Hilton, F. Hirzebruch, and R. Remmert). Berlin: Springer-Verlag, pp. 7-44, 1991.Beslin, S. and de Angelis, V. "The Minimal Polynomials of sin((2pi)/p) and cos((2pi)/p)." Math. Mag. 77, 146-149, 2004.Carrega, J. C. Théorie des corps. Paris: Hermann, 1989.Cigler, J. Körper, Ringe, Gleichungen. Heidelberg, Germany: Spektrum, 1995.Conway, J. H. and Guy, R. K. In The Book of Numbers. New York: Springer-Verlag, 1996.Dickson, L. E. Modern Algebraic Theories. Chicago: H. Sanborn, 1926.Dörrie, H. Quadratische Gleichungen. Munich: Oldenburg, 1943.Fricke, R. Lehrbuch der Algebra, 2 vols. Braunschweig, Germany: Vieweg, 1926.Gardner, M. "Pascal's Triangle." Ch. 15 in Mathematical Carnival: A New Round-Up of Tantalizers and Puzzles from Scientific American. New York: Vintage Books, pp. 194-207, 1977.Hobson, E. W. Squaring the Circle. New York: Chelsea, 1988.Jones, A.; Morris, S. A.; and Pearson, K. R. Abstract Algebra and Famous Impossibilities. New York: Springer-Verlag, 1991.Josephy, M. "An Afterthought of Gauss on Cyclotomy." In Proceedings of the 2nd Gauss Symposium. Conference A: Mathematics and Theoretical Physics. Held at Ludwig-Maximilians University, Munich, August 2-7, 1993 (Ed. M. Behara, R. Fritsch, and R. G. Lintz). Berlin: de Gruyter, pp. 147-150, 1995.Kommerell, K. Das Grenzgebiet der elementaren und hoeheren Mathematik. Leipzig, Germany: Koehler, 1936.Krížek, M.; Luca, F.; and Somer, L. 17 Lectures on Fermat Numbers: From Number Theory to Geometry. New York: Springer-Verlag, 2001.Lehmer, D. H. "A Note on Trigonometric Algebraic Numbers." Amer. Math. Monthly 40, 165-166, 1933.Mitzscherling, A. Das Problem der Kreisteilung. Leipzig, Germany: Teubner, 1913.Mora, T. Solving Polynomial Equation Systems, Vol. 1. Cambridge, England: Cambridge University Press, 2003.Netto, E. Vorlesungen über Algebra, Vols. 1-2. Leipzig, Germany: Teubner, 1896.Niven, I. M. Irrational Numbers. New York: Wiley, p. 39, 1956.Ribenboim, P. Algebraic Numbers. New York: Wiley, 1972.Sloane, N. J. A. Sequences A000265/M2222, A003401/M0505, A023022, A055034, A055035, A084623, A093819, A113401, and A113402 in "The On-Line Encyclopedia of Integer Sequences."Smith, D. E. A Source Book in Mathematics. New York: Dover, 1994.Stewart, I. Galois Theory, 3rd ed. London: Chapman and Hall, London, 1972.Strommer, J. "Konstruktion des regulären 257-Ecks mit Lineal und Streckenübertrager." Acta Math. Hungar. 70, 259-292, 1996.Surowski, D. and McCombs, P. "Homogeneous Polynomials and the Minimal Polynomial ofcos(2pi/n)." Missouri J. Math. Sci. 15, 4-14, 2003.Tietze, H. Famous Problems of Mathematics: Solved and Unsolved Mathematics Problems from Antiquity to Modern Times. New York: Graylock Press, 1965.Trott, M. "cos(2pi/257) à la Gauss." Mathematica Educ. Res. 4, 31-36, 1995.Trott, M. "cos(2pi/257) à la Gauss." §1.10.2 in The Mathematica GuideBook for Symbolics. New York: Springer-Verlag, pp. 312-321, 2006.élez, P. and Luis, O. "A Chord Approach for an Alternative Ruler and Compasses Construction of the 17-Side Regular Polygon." Geom. Dedicata 52, 209-213, 1994.Wantzel, M. L. "Recherches sur les moyens de reconnaître si un Problème de Géométrie peut se résoudre avec la règle et le compas." J. Math. pures appliq. 1, 366-372, 1836.Watkins, W. and Zeitlin, J. "The Minimal Polynomial of cos(2pi/n)." Amer. Math. Monthly 100, 471-474, 1993.Weber, A. "Computing Radical Expressions for Roots of Unity." SIGSAM Bull. 30, 11-20, 1996.Wegner, K. W. "Equations with Trigonometric Values as Roots." Amer. Math. Monthly 66, 52-53, 1959.

Referenced on Wolfram|Alpha

Trigonometry Angles

Cite this as:

Weisstein, Eric W. "Trigonometry Angles." From MathWorld--A Wolfram Web Resource.

Subject classifications