Sine Function is Odd

Theorem
Let $x \in \R$ be a real number.

Let $\sin x$ be the sine of $x$.

Then:
 * $\sin \left({-x}\right) = -\sin x$

That is, the sine function is odd.

Proof
Recall the definition of the sine function:


 * $\displaystyle \sin x = \sum_{n \mathop = 0}^\infty \left({-1}\right)^n \frac {x^{2n+1}}{\left({2n+1}\right)!} = x - \frac {x^3} {3!} + \frac {x^5} {5!} - \cdots$

From Sign of Odd Power, we have that:
 * $\forall n \in \N: -\left({x^{2n+1}}\right) = \left({-x}\right)^{2n+1}$

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