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: \begin{eqnarray*} S^2 (r) + C^2 (r) &=& 1 \cr S (p + q) &=& S(p) C(q) + S(q) C(p) \cr C (p + q) &=& C(p) C(q) - S(p) S(q) \end{eqnarray*} 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, $\sin(\frac{\pi}{2})= 1$ .