Euler Formula for Sine Function/Real Numbers/Proof 3

Theorem

 * $\displaystyle \sin x = x \prod_{n \mathop = 1}^\infty \left({1 - \frac {x^2} {n^2 \pi^2}}\right)$

for all $x \in \R$.

Proof
Euler's use of complex numbers can be avoided as follows.

For odd $n$, we have that $\sin x$ is a polynomial of degree $n$ in $\sin \dfrac x n$.

The roots of this polynomial are the numbers $\sin \dfrac {k \pi} n$ where $k$ is any integer.

The result follows from:


 * Factoring the polynomial
 * making $n$ go to infinity
 * replacing $\sin y$ by $y$ for small $y$.