Trigonometry/Power Series for Cosine and Sine

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

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

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

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

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

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

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