# rigorous definition of trigonometric functions

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 RigorousDefinitionOfTrigonometricFunctions 2013-03-22 16:22:11 2013-03-22 16:22:11 CWoo (3771) CWoo (3771) 9 CWoo (3771) Derivation msc 26A09 TrigonometricFormulasFromSeries