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:

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

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.

$\blacksquare$

Sources

• Murray R. Spiegel: Mathematical Handbook of Formulas and Tables (1968): $5.28$
• K.G. Binmore: Mathematical Analysis: A Straightforward Approach (1977)... (previous)... (next): $\S 16.3 \ (1) \ \text{(iv)}$