Sine Function is Odd

Theorem
For all $z \in \C$:
 * $\sin \paren {-z} = -\sin z$

That is, the sine function is odd.

Proof
Recall the definition of the sine function:


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

From Sign of Odd Power, we have that:
 * $\forall n \in \N: -\paren {z^{2 n + 1} } = \paren {-z}^{2 n + 1}$

The result follows directly.

Also see

 * Cosine Function is Even
 * Tangent Function is Odd
 * Cotangent Function is Odd
 * Secant Function is Even
 * Cosecant Function is Odd


 * Hyperbolic Sine Function is Odd