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
We have that $\sin x$ has a power series representation:


 * $\displaystyle x - \frac {x^3} {3!} + \frac {x^5} {5!} - \frac{x^7}{7!} + \cdots$

The roots of sine are the numbers $k \pi$, where $k$ is any integer.

From the Polynomial Factor Theorem, the following might be true:


 * $\displaystyle \sin x = Ax \prod \left({1 - \frac x { k\pi }}\right)$

where the product is taken over all $n \in \Z \setminus \left\{{0}\right\}$, and $A$ is some constant.

The intuition is as follows.

That $\dfrac {\sin x}x \to 1$ as $x \to 0$ is a well known limit. Letting $x$ tend to $0$ in the above equation implies that $A = 1$.

We now formalize the above claims.