rigorous definition of trigonometric functions

It is possible to define the trigonometric functions rigorously by means of aprocess based upon the angle addition identities. A sketch of how this is doneis 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 realnumbers 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 $n\\to\\infty$ , itfollows that, if $\\{p_{n}\\}_{n=1}^{\\infty}$ and $\\{q_{n}\\}_{n=1}^{\\infty}$ aretwo sequences of rational numbers whose denominators are powers of twosuch 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 convergentseries of rational numbers $\\{r_{n}\\}_{n=0}^{\\infty}$ whose denominatorsare 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 additionidentities.

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, $\\sin(\\frac{\\pi}{2})=1$ .