Euler Formula for Sine Function/Real Numbers/Proof 2

Theorem

\(\ds \sin x\)

\(=\)

\(\ds x \prod_{n \mathop = 1}^\infty \paren {1 - \frac {x^2} {n^2 \pi^2} }\)

\(\ds \)

\(=\)

\(\ds x \paren {1 - \dfrac {x^2} {\pi^2} } \paren {1 - \dfrac {x^2} {4 \pi^2} } \paren {1 - \dfrac {x^2} {9 \pi^2} } \dotsm\)

for all $x \in \R$.

Proof

Using De Moivre's Formula:

$\sin x = \dfrac {\left({\cos \dfrac x n + i \sin \dfrac x n}\right)^n - \left({\cos \dfrac x n - i \sin \dfrac x n}\right)^n} {2i}$

The difference between two $n$th powers can be extracted into linear factors using $n$th roots of unity.

For large $n$, we can replace:

$\cos \dfrac x n$ by $1$

$\sin \dfrac x n$ by $\dfrac x n$

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Source of Name

This entry was named for Leonhard Paul Euler.

He proved it in vol. 1 of his 1748 work Introductio in analysin infinitorum using De Moivre's Formula as above.