# Euler Formula for Sine Function/Real Numbers/Proof 3

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## Theorem

- $\displaystyle \sin x = x \prod_{n \mathop = 1}^\infty \paren {1 - \frac {x^2} {n^2 \pi^2} }$

for all $x \in \R$.

## Proof

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

- $\sin x = x - \dfrac {x^3} {3!} + \dfrac {x^5} {5!} - \dfrac {x^7} {7!} + \cdots$

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

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

- $\displaystyle \sin x = A x \prod \paren {1 - \frac x {k \pi} }$

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

The intuition is as follows.

\(\displaystyle \sin x\) | \(=\) | \(\displaystyle \ldots \paren {1 - \frac x {2 \pi} } \paren {1 - \frac x \pi} A x \paren {1 + \frac x \pi} \paren {1 + \frac x {2 \pi} } \cdots\) | |||||||||||

\(\displaystyle \) | \(=\) | \(\displaystyle A x \paren {1 - \frac {x^2} {\pi^2} } \paren {1 - \frac {x^2} {2^2 \pi^2} } \paren {1 - \frac {x^2} {3^2 \pi^2} } \cdots\) | |||||||||||

\(\displaystyle \leadsto \ \ \) | \(\displaystyle \frac {\sin x} x\) | \(=\) | \(\displaystyle A \paren {1 - \frac {x^2} {\pi^2} } \paren {1 - \frac {x^2} {2^2 \pi^2} } \paren {1 - \frac {x^2} {3^2 \pi^2} } \cdots\) | for $x \ne 0$ |

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.

## Sources

- 1992: George F. Simmons:
*Calculus Gems*... (previous) ... (next): Chapter $\text {B}.14$: Euler's Discovery of the Formula $\displaystyle \sum_1^\infty \frac 1 {n^2} = \frac {\pi^2} 6$