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rigorous definition of trigonometric
functions∗
CWoo†
2013-03-21 21:24:40
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
r
cn
cn+1 = 1 − 1 − .
2
Likewise define a sequence {sn }∞
n=1 by the conditions s1 = 1 and
r
cn
.
sn+1 =
2
(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:
1 − cn
sn
mn =
−sn
1 − cn
Using the recursion relations which define cn and sn , it may be shown that
k
m2n+1 = mn , More grenerally, using induction, this can be generalised to m2n+k =
mn .
It is easy to check that the product of any two matrices of the form
x y
−y x
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
∗ hRigorousDefinitionOfTrigonometricFunctionsi created: h2013-03-21i by: hCWooi version: h38508i Privacy setting: h1i hDerivationi h26A09i
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whose denominator is a power of two to real numbers by the following equation:
n
C 2nk S 2nk 1 − ck
sk
=
.
−sk
1 − ck
−S 2nk
C 2nk
From the recursion relations, we may prove the following identities:
S 2 (r) + C 2 (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 π lim rn = lim C(rn )
n→∞
and
n→∞
sin π lim rn = lim S(rn ).
n→∞
n→∞
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 := 21 for every positive integer i. Then we need to find C( 21 ) and S( 12 ).
We use the matrix above defining C and S, and set n = k = 1:
C 12 S 21 1 − c1
s1
0 1
=
=
.
−s1
1 − c1
−1 0
−S 12
C 12
As a result, cos( π2 ) = cos(π limi→∞ 12 ) = limi→∞ C( 21 ) = C( 12 ) = 0. Similarly,
sin( π2 ) = 1.
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