Sine of Zero is Zero

Theorem

$\sin 0 = 0$

where $\sin$ denotes the sine.

Proof

Recall the definition of the sine function:

$\displaystyle \sin x = \sum_{n \mathop = 0}^\infty \paren {-1}^n \frac {x^{2 n + 1} } {\paren {2 n + 1}!} = x - \frac {x^3} {3!} + \frac {x^5} {5!} - \cdots$

Thus:

$\displaystyle \sin 0 = 0 - \frac {0^3} {3!} + \frac {0^5} {5!} - \cdots = 0$

$\blacksquare$