rigorous definition of trigonometric functions
It is possible to define the trigonometric functions 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 sequence {cn}∞n=1 by the initial condition
c1=1 and the recursion
cn+1=1-√1-cn2. |
Likewise define a sequence {sn}∞n=1 by the conditions s1=1 and
sn+1=√cn2. |
(In both equations above, we take the positive square root.)
It may be shown that both of these sequences are strictly decreasing
and approach 0.
Next, define a sequence of 2×2 matrices as follows:
mn=(1-cnsn-sn1-cn) |
Using the recursion relations which define cn and sn, it may be shown that m2n+1=mn, More grenerally, using induction, this can be generalised to m2kn+k=mn.
It is easy to check that the product of any two matrices of the form
(xy-yx) |
is of the same form. Hence, for any integers k and n, the matrix
mkn will be of this form. We can therefore define functions S and
C from rational numbers whose denominator is a power of two to real
numbers by the following equation:
(C(n2k)S(n2k)-S(n2k)C(n2k))=(1-cksk-sk1-ck)n. |
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 cn→0 and sn→0 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 limn→∞pn=limn→∞qn, then
limn→∞C(pn)=limn→∞C(qn) and
limn→∞S(pn)=limn→∞S(qn). Therefore,
we may define functions by the conditions that, for any convergent
series of rational numbers {rn}∞n=0 whose denominators
are powers of two,
cos(πlimn→∞rn)=limn→∞C(rn) |
and
sin(πlimn→∞rn)=limn→∞S(rn). |
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:
(C(12)S(12)-S(12)C(12))=(1-c1s1-s11-c1)=(01-10). |
As a result, cos(π2)=cos(πlimi→∞12)=limi→∞C(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 |