Trigonometric constant expressed in real radicals

The primary solution angles in the form (cos, sin) on the unit circle are multiples of 30 ° and 45 °.

There are infinitely many angles for which the values ​​of the trigonometric functions can be given by explicit exact expressions using elementary arithmetic operations and radical expressions . In the following only square roots are considered. If we know this expression for the sine , we can determine this for the cosine (and vice versa) with the trigonometric identity . In this case we will definitely find an exact value for the tangent and cotangent . ${\displaystyle \sin ^{2}x+\cos ^{2}x=1}$

${\displaystyle \cos \left({\frac {2\pi }{n}}\right)}$can be expressed with real square roots if and only if the regular corner is a constructible polygon : This means in concrete terms that a product of a power of 2 and pairwise different Fermat prime numbers is. ${\displaystyle n}$ ${\displaystyle n}$

The sine and cosine of the angles, which are multiples of 3 °, can be expressed in an elementary way using real square roots. For angles that are a multiple of 1 ° but not a multiple of 3 °, on the other hand, it is not possible to find an expression with real square roots. This is due to the fact that such expressions with real square roots correspond exactly to the angles that can be constructed with compasses and rulers and that in the prefactor (the angles were given in degrees) in the denominator one appears. One of the Fermat prime numbers in the construction of regular n-vertices with compasses and ruler is the one that may only appear once as a factor of in . ${\displaystyle {\frac {2\pi }{360}}={\frac {2\pi }{9\cdot 8\cdot 5}}}$ ${\displaystyle 9}$ ${\displaystyle 3}$ ${\displaystyle n}$ ${\displaystyle \cos \left({\frac {2\pi }{n}}\right)}$

If we know an exact real radical expression for the sine and cosine of an angle, then we can determine the exact half-angle real radical expressions using the half-angle formulas. This means that there are an infinite number of angles between 0 ° and 360 ° for which such expressions can be generated with real radicals.

In general it can be shown that the values ​​of cosine and sine (and other trigonometric functions formed from them such as tangent, cotangent) in rational multiples of (if the argument is a rational number in degrees) are algebraic numbers . The proof is simple, uses Euler's formula , De Moivre's theorem and the separation into real and imaginary parts. Conversely, according to a theorem of Alan Baker , if the values ​​of sine or cosine are algebraic at any point , it is either rational or transcendent (but not algebraic and irrational). Like the other trigonometric functions, cosine and sine themselves are transcendent functions , which means that they do not satisfy any algebraic equations. The question of when sine and cosine have rational values ​​in rational places is answered by Niven's theorem. ${\displaystyle \pi }$ ${\displaystyle a2\pi }$ ${\displaystyle a}$

Exact values ​​for multiples of 3 °

The sine and cosine of angles that are multiples of 3 ° ( ) can be expressed in real radicals. Due to the elementary properties of the sine and the cosine, we limit ourselves in the following to angles between 0 ° and 45 °. ${\displaystyle \textstyle {\mathrel {\hat {=}}}{\frac {2\pi }{120}}}$

${\displaystyle {\boldsymbol {\theta }}}$	${\displaystyle {\boldsymbol {\cos \theta }}}$	${\displaystyle {\boldsymbol {\sin \theta }}}$
0 °	${\displaystyle 1}$	${\displaystyle 0}$
3 °	${\displaystyle {\frac {2({\sqrt {3}}+1){\sqrt {5+{\sqrt {5}}}}+{\sqrt {2}}({\sqrt {3}}-1)({\sqrt {5}}-1)}{16}}}$	${\displaystyle {\frac {{\sqrt {2}}({\sqrt {3}}+1)({\sqrt {5}}-1)-2({\sqrt {3}}-1){\sqrt {5+{\sqrt {5}}}}}{16}}}$
6 °	${\displaystyle {\frac {{\sqrt {2}}{\sqrt {5-{\sqrt {5}}}}+{\sqrt {3}}({\sqrt {5}}+1)}{8}}}$	${\displaystyle {\frac {{\sqrt {6}}{\sqrt {5-{\sqrt {5}}}}-({\sqrt {5}}+1)}{8}}}$
9 °	${\displaystyle {\frac {{\sqrt {2}}({\sqrt {5}}+1)+({\sqrt {5}}-1){\sqrt {5+{\sqrt {5}}}}}{8}}}$	${\displaystyle {\frac {{\sqrt {2}}({\sqrt {5}}+1)-({\sqrt {5}}-1){\sqrt {5+{\sqrt {5}}}}}{8}}}$
12 °	${\displaystyle {\frac {{\sqrt {6}}{\sqrt {5+{\sqrt {5}}}}+{\sqrt {5}}-1}{8}}}$	${\displaystyle {\frac {{\sqrt {2}}{\sqrt {5+{\sqrt {5}}}}-{\sqrt {3}}({\sqrt {5}}-1)}{8}}}$
15 °	${\displaystyle {\frac {\sqrt {2+{\sqrt {3}}}}{2}}}$	${\displaystyle {\frac {\sqrt {2-{\sqrt {3}}}}{2}}}$
18 °	${\displaystyle {\frac {{\sqrt {2}}{\sqrt {5+{\sqrt {5}}}}}{4}}}$	${\displaystyle {\frac {{\sqrt {5}}-1}{4}}}$
21 °	${\displaystyle {\frac {2({\sqrt {3}}-1){\sqrt {5-{\sqrt {5}}}}+{\sqrt {2}}({\sqrt {3}}+1)({\sqrt {5}}+1)}{16}}}$	${\displaystyle {\frac {2({\sqrt {3}}+1){\sqrt {5-{\sqrt {5}}}}-{\sqrt {2}}({\sqrt {3}}-1)({\sqrt {5}}+1)}{16}}}$
24 °	${\displaystyle {\frac {{\sqrt {5}}+1+{\sqrt {6}}{\sqrt {5-{\sqrt {5}}}}}{8}}}$	${\displaystyle {\frac {{\sqrt {3}}({\sqrt {5}}+1)-{\sqrt {2}}{\sqrt {5-{\sqrt {5}}}}}{8}}}$
27 °	${\displaystyle {\frac {2{\sqrt {5+{\sqrt {5}}}}+{\sqrt {2}}({\sqrt {5}}-1)}{8}}}$	${\displaystyle {\frac {2{\sqrt {5+{\sqrt {5}}}}-{\sqrt {2}}({\sqrt {5}}-1)}{8}}}$
30 °	${\displaystyle {\frac {\sqrt {3}}{2}}}$	${\displaystyle {\frac {1}{2}}}$
33 °	${\displaystyle {\frac {2({\sqrt {3}}+1){\sqrt {5+{\sqrt {5}}}}-{\sqrt {2}}({\sqrt {3}}-1)({\sqrt {5}}-1)}{16}}}$	${\displaystyle {\frac {2({\sqrt {3}}-1){\sqrt {5+{\sqrt {5}}}}+{\sqrt {2}}({\sqrt {3}}+1)({\sqrt {5}}-1)}{16}}}$
36 °	${\displaystyle {\frac {{\sqrt {5}}+1}{4}}}$	${\displaystyle {\frac {{\sqrt {2}}{\sqrt {5-{\sqrt {5}}}}}{4}}}$
39 °	${\displaystyle {\frac {{\sqrt {2}}({\sqrt {3}}-1)({\sqrt {5}}+1)+2({\sqrt {3}}+1){\sqrt {5-{\sqrt {5}}}}}{16}}}$	${\displaystyle {\frac {{\sqrt {2}}({\sqrt {3}}+1)({\sqrt {5}}+1)-2({\sqrt {3}}-1){\sqrt {5-{\sqrt {5}}}}}{16}}}$
42 °	${\displaystyle {\frac {{\sqrt {2}}{\sqrt {5+{\sqrt {5}}}}+{\sqrt {3}}({\sqrt {5}}-1)}{8}}}$	${\displaystyle {\frac {{\sqrt {6}}{\sqrt {5+{\sqrt {5}}}}-({\sqrt {5}}-1)}{8}}}$
45 °	${\displaystyle {\frac {\sqrt {2}}{2}}}$	${\displaystyle {\frac {\sqrt {2}}{2}}}$

To determine the values ​​in the table, you can proceed as follows: The 0 ° angle is trivial; the 30 ° and 45 ° angles can be derived from the corresponding triangles, which are each half of an isosceles triangle and a square; for the 18 ° angle we can use the fact that the ratio between a diagonal and a side in a regular pentagon corresponds to the golden ratio . All other values ​​in the table can be determined using these angles and the trigonometric identities such as addition, subtraction, half angles and double angles.

Exact expressions derived from constructible polygons

The following table shows the first values ​​of and , which are expressed in real radicals. It should be noted that a product of a power of 2 and mutually different Fermat prime numbers , or, equivalent, that the regular n-gon must be constructible. The only known Fermat primes are 3, 5, 17, 257 and 65537. ${\displaystyle \sin \left({\frac {2\pi }{n}}\right)}$ ${\displaystyle \cos \left({\frac {2\pi }{n}}\right)}$ ${\displaystyle n}$

${\displaystyle {\textbf {n}}}$	${\displaystyle \cos \left({\frac {2\pi }{n}}\right)}$	${\displaystyle \sin \left({\frac {2\pi }{n}}\right)}$
${\displaystyle 1}$	${\displaystyle 1}$	${\displaystyle 0}$
${\displaystyle 2}$	${\displaystyle -1}$	${\displaystyle 0}$
${\displaystyle 3}$	${\displaystyle -{\frac {1}{2}}}$	${\displaystyle {\frac {\sqrt {3}}{2}}}$
${\displaystyle 4}$	${\displaystyle 0}$	${\displaystyle 1}$
${\displaystyle 5}$	${\displaystyle {\frac {{\sqrt {5}}-1}{4}}}$	${\displaystyle {\frac {{\sqrt {2}}{\sqrt {5+{\sqrt {5}}}}}{4}}}$
${\displaystyle 6}$	${\displaystyle {\frac {1}{2}}}$	${\displaystyle {\frac {\sqrt {3}}{2}}}$
${\displaystyle 8}$	${\displaystyle {\frac {\sqrt {2}}{2}}}$	${\displaystyle {\frac {\sqrt {2}}{2}}}$
${\displaystyle 10}$	${\displaystyle {\frac {{\sqrt {5}}+1}{4}}}$	${\displaystyle {\frac {{\sqrt {2}}{\sqrt {5-{\sqrt {5}}}}}{4}}}$
${\displaystyle 12}$	${\displaystyle {\frac {\sqrt {3}}{2}}}$	${\displaystyle {\frac {1}{2}}}$
${\displaystyle 16}$	${\displaystyle {\frac {\sqrt {2+{\sqrt {2}}}}{2}}}$	${\displaystyle {\frac {\sqrt {2-{\sqrt {2}}}}{2}}}$

Theorem of nives

According to Niven's theorem , 30 ° is the only angle of degrees that is a rational number in degrees and the sine of which is also a rational number. The only cases in which the sine or cosine is rational for a rational value of the angle in degrees are thus in the first quadrant 0, 30, 60 and 90 degrees. The only rational values ​​that sine and cosine take for rational arguments in degrees are . The same can be shown for the tangent, the only rational values ​​for rational values ​​of the angle in degrees are here and (and also with the cotangent). ${\displaystyle 0<\theta <90}$ ${\displaystyle 0,\pm 1,\pm {\frac {1}{2}}}$ ${\displaystyle 0}$ ${\displaystyle \pm 1}$

See also

• Sine and cosine
• Trigonometry formula collection
• Constructible polygon

literature

• Brian Bradie: Exact values ​​for the sine and cosine of multiples of 18 °: A geometric approach . The College Mathematics Journal. Volume 33, 2002, pp. 318-319. doi: 10.2307 / 1559057 .
• John Horton Conway , Charles Radin , Lorenzo Radun: On angles whose squared trigonometric functions are rational . Disc. And Comp. Geom., Vol. 22, 1999, pp. 321-332, Arxiv
• Kurt Girstmair: Some linear relations between values ​​of trigonometric functions at kπ / n . Acta Arithmetica, Vol. 81, 1997, pp. 387-398. MR 1472818.
• S. Gurak: On the minimal polynomial of gauss periods for prime powers , Mathematics of Computation, Volume 75, 2006, pp. 2021-2035. Bibcode: 2006MaCom..75.2021G. doi: 10.1090 / S0025-5718-06-01885-0 . MR 2240647.
• LD Servi: Nested square roots of 2 , Amer. Math. Monthly, Vol. 110, 2003, pp. 326-330. doi: 10.2307 / 3647881 . JSTOR 3647881. MR 1984573.

Individual evidence

1. ↑ John Horton Conway and Richard K. Guy name such expressions, i.e. algebraic numbers that correspond to geometrical constructions with compasses and ruler, Euclidean numbers , Conway, Guy, The Book of Numbers, 1996, p. 192
2. ↑ David Richeson, division by zero , blog
3. ^ Niven's theorem , Mathworld. Ivan Niven , Irrational Numbers, Carus Mathematical Monographs, Wiley 1956, p. 41