trigonometric formulas from series

One may define the sine and the cosine functions for real (and complex) arguments using the power seriesMathworldPlanetmath

sinx=x-x33!+x55!-+, (1)
cosx= 1-x22!+x44!-+, (2)

and using only the properties of power series, easily derive most of the goniometric formulasPlanetmathPlanetmath, without any geometry.  For example, one gets instantly from (1) and (2) the values

sin0= 0,cos0= 1

and the parity relations (


Using the Cauchy multiplication rule for series one can obtain the addition formulasPlanetmathPlanetmath

{sin(x+y)=sinxcosy+cosxsiny,cos(x+y)=cosxcosy-sinxsiny. (3)

These produce straightforward many other important formulae, e.g.

sin2x= 2sinxcosx,cos2x=cos2x-sin2x  (y=:x) (4)


cos2x+sin2x= 1      (y=:-x). (5)

The value  cosπ2=0,  as well as the formulae expressing the periodicity of sine and cosine, cannot be directly obtained from the series (1) and (2) — in fact, one must define the number π by using the functionMathworldPlanetmath properties of the and its derivativeMathworldPlanetmath series (

The equation

cosx= 0

has on the interval(0, 2)  exactly one root (  Actually, as sum of a power series, cosx is continuousMathworldPlanetmath,  cos0=1>0  and  cos2<1-222!+244!<0  (see Leibniz’ estimate for alternating seriesMathworldPlanetmath (, whence there is at least one root.  If there were more than one root, then the derivative


would have at least one zero on the interval; this is impossible, since by Leibniz the series in the parentheses does not change its sign on the interval:

1-x23!+-> 1-223!> 0

Accordingly, we can define the number pi to be the least positive solution of the equation  cosx=0, multiplied by 2.

Thus we have  0<π<4  and  cosπ2=0.  Furthermore, by (5),

sinπ2= 1,

and by (4),

sinπ= 0,cosπ=-1,sin2π= 0,cos2π= 1.

Consequently, the addition formulas (3) yield the periodicities (

Title trigonometric formulas from series
Canonical name TrigonometricFormulasFromSeries
Date of creation 2013-03-22 18:50:47
Last modified on 2013-03-22 18:50:47
Owner pahio (2872)
Last modified by pahio (2872)
Numerical id 9
Author pahio (2872)
Entry type Derivation
Classification msc 26A09
Synonym series definition of sine and cosine
Related topic RigorousDefinitionOfTrigonometricFunctions
Related topic ApplicationOfFundamentalTheoremOfIntegralCalculus
Related topic TrigonometricFormulasFromDeMoivreIdentity
Related topic GoniometricFormulae
Defines π