algebraic sines and cosines

For any rational numberPlanetmathPlanetmathPlanetmath r, the sine and the cosine of the number rπ are algebraic numbersMathworldPlanetmath.

Proof.  According to the entry, sinnφ and cosnφ can be expressed as polynomialsPlanetmathPlanetmath with integer coefficients of sinφ or cosφ, respectively, when n is an integer.  Thus we can write


where  P(x),Q(x)[x].  If  r=mn  where m,n are integers and  n0,  we have

P(sinrπ)=sinnrπ=sinmπ= 0,Q(cosrπ)=cosnrπ=cosmπ=±1,

i.e. both sinrπ and cosrπ satisfy an algebraic equation.  Q.E.D.

For example,

cos7φ= 64cos7φ-112cos5φ+56cos3φ-7cosφ,

whence we have the identity

64cos7π7-112cos5π7+56cos3π7-7cosπ7+1= 0,

and therefore cosπ7 is algebraic over .

Title algebraic sines and cosines
Canonical name AlgebraicSinesAndCosines
Date of creation 2013-03-22 18:51:27
Last modified on 2013-03-22 18:51:27
Owner pahio (2872)
Last modified by pahio (2872)
Numerical id 7
Author pahio (2872)
Entry type Corollary
Classification msc 11R04
Classification msc 11C08
Related topic RationalSineAndCosine
Related topic MultiplesOfAnAlgebraicNumber