rigorous definition of trigonometric functions

It is possible to define the trigonometric functionsDlmfMathworldPlanetmath rigorously by means of a process based upon the angle addition identities. A sketch of how this is done is provided below.

To begin, define a sequenceMathworldPlanetmath {cn}n=1 by the initial conditionMathworldPlanetmath c1=1 and the recursion


Likewise define a sequence {sn}n=1 by the conditions s1=1 and


(In both equations above, we take the positive square rootMathworldPlanetmath.) It may be shown that both of these sequences are strictly decreasingPlanetmathPlanetmath and approach 0.

Next, define a sequence of 2×2 matrices as follows:


Using the recursion relations which define cn and sn, it may be shown that mn+12=mn, More grenerally, using induction, this can be generalised to mn+k2k=mn.

It is easy to check that the product of any two matrices of the form


is of the same form. Hence, for any integers k and n, the matrix mnk will be of this form. We can therefore define functionsMathworldPlanetmath S and C from rational numbers whose denominator is a power of two to real numbers by the following equation:


From the recursion relations, we may prove the following identities:

S2(r)+C2(r) = 1
S(p+q) = S(p)C(q)+S(q)C(p)
C(p+q) = C(p)C(q)-S(p)S(q)

From the fact that cn0 and sn0 as n, it follows that, if {pn}n=1 and {qn}n=1 are two sequences of rational numbers whose denominators are powers of two such that limnpn=limnqn, then limnC(pn)=limnC(qn) and limnS(pn)=limnS(qn). Therefore, we may define functions by the conditions that, for any convergent seriesMathworldPlanetmathPlanetmath of rational numbers {rn}n=0 whose denominators are powers of two,




By continuity, we see that these functions satisfy the angle addition identities.

Application. Let us use the definitions above to find sin(π2) and cos(π2). Let ri:=12 for every positive integer i. Then we need to find C(12) and S(12). We use the matrix above defining C and S, and set n=k=1:


As a result, cos(π2)=cos(πlimi12)=limiC(12)=C(12)=0. Similarly, sin(π2)=1.

Title rigorous definition of trigonometric functions
Canonical name RigorousDefinitionOfTrigonometricFunctions
Date of creation 2013-03-22 16:22:11
Last modified on 2013-03-22 16:22:11
Owner CWoo (3771)
Last modified by CWoo (3771)
Numerical id 9
Author CWoo (3771)
Entry type Derivation
Classification msc 26A09
Related topic TrigonometricFormulasFromSeries