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 $\{c_{n}\}_{n=1}^{\infty}$ by the initial condition $c_{1}=1$ and the recursion

$c_{n+1}=1-\sqrt{1-{c_{n}\over 2}}.$

Likewise define a sequence $\{s_{n}\}_{n=1}^{\infty}$ by the conditions $s_{1}=1$ and

$s_{n+1}=\sqrt{c_{n}\over 2}.$

(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 $2\times 2$ matrices as follows:

$m_{n}=\left(\begin{matrix}1-c_{n}&s_{n}\\ -s_{n}&1-c_{n}\end{matrix}\right)$

Using the recursion relations which define $c_{n}$ and $s_{n}$ , it may be shown that $m_{n+1}^{2}=m_{n}$ , More grenerally, using induction, this can be generalised to $m_{n+k}^{2^{k}}=m_{n}$ .

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

$\left(\begin{matrix}x&y\\ -y&x\end{matrix}\right)$

is of the same form. Hence, for any integers $k$ and $n$ , the matrix $m_{n}^{k}$ 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:

$\left(\begin{matrix}C\left({n\over 2^{k}}\right)&S\left({n\over 2^{k}}\right)% \\ -S\left({n\over 2^{k}}\right)&C\left({n\over 2^{k}}\right)\end{matrix}\right)=% \left(\begin{matrix}1-c_{k}&s_{k}\\ -s_{k}&1-c_{k}\end{matrix}\right)^{n}.$

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

$\displaystyle S^{2}(r)+C^{2}(r)$	$\displaystyle=$	$\displaystyle 1$
$\displaystyle S(p+q)$	$\displaystyle=$	$\displaystyle S(p)C(q)+S(q)C(p)$
$\displaystyle C(p+q)$	$\displaystyle=$	$\displaystyle C(p)C(q)-S(p)S(q)$

From the fact that $c_{n}\to 0$ and $s_{n}\to 0$ as $n\to\infty$ , it follows that, if $\{p_{n}\}_{n=1}^{\infty}$ and $\{q_{n}\}_{n=1}^{\infty}$ are two sequences of rational numbers whose denominators are powers of two such that $\lim_{n\to\infty}p_{n}=\lim_{n\to\infty}q_{n}$ , then $\lim_{n\to\infty}C(p_{n})=\lim_{n\to\infty}C(q_{n})$ and $\lim_{n\to\infty}S(p_{n})=\lim_{n\to\infty}S(q_{n})$ . Therefore, we may define functions by the conditions that, for any convergent series of rational numbers $\{r_{n}\}_{n=0}^{\infty}$ whose denominators are powers of two,

$\cos\left(\pi\lim_{n\to\infty}r_{n}\right)=\lim_{n\to\infty}C(r_{n})$

and

$\sin\left(\pi\lim_{n\to\infty}r_{n}\right)=\lim_{n\to\infty}S(r_{n}).$

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

Application. Let us use the definitions above to find $\sin(\frac{\pi}{2})$ and $\cos(\frac{\pi}{2})$ . Let $r_{i}:=\frac{1}{2}$ for every positive integer $i$ . Then we need to find $C(\frac{1}{2})$ and $S(\frac{1}{2})$ . We use the matrix above defining $C$ and $S$ , and set $n=k=1$ :

$\left(\begin{matrix}C\left({\frac{1}{2}}\right)&S\left({\frac{1}{2}}\right)\\ -S\left({\frac{1}{2}}\right)&C\left({\frac{1}{2}}\right)\end{matrix}\right)=% \left(\begin{matrix}1-c_{1}&s_{1}\\ -s_{1}&1-c_{1}\end{matrix}\right)=\left(\begin{matrix}0&1\\ -1&0\end{matrix}\right).$

As a result, $\cos(\frac{\pi}{2})=\cos(\pi\lim_{i\to\infty}\frac{1}{2})=\lim_{i\to\infty}C(% \frac{1}{2})=C(\frac{1}{2})=0$ . Similarly, $\sin(\frac{\pi}{2})=1$ .

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