Trigonometry/Power Series for Cosine and Sine

Applying Maclaurin's theorem to the cosine and sine functions for angle x (in radians), we get


 * $$\displaystyle \cos(x) = 1 - {x^{2} \over 2!} + {x^{4} \over 4!} - \cdots = \sum_{n=0}^{\infty} \frac{(-1)^nx^{2n}}{(2n)!}$$


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

For both series, the ratio of the $$n^{th}$$ to the $$(n-1)^{th}$$ term tends to zero for all $$x$$. Thus, both series are absolutely convergent for all $$x$$.

Many properties of the cosine and sine functions can easily be derived from these expansions, such as


 * $$\displaystyle \sin(-x) = -\sin(x)$$


 * $$\displaystyle \cos(-x) = \cos(x)$$


 * $$\displaystyle \frac{d}{dx} \sin(x) = \cos(x)$$