View source for Trigonometry/Power Series for Cosine and Sine

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

:<math>\displaystyle \cos(x) = 1 - {x^{2} \over 2!} + {x^{4} \over 4!} - \cdots = \sum_{n=0}^{\infty} \frac{(-1)^nx^{2n}}{(2n)!}</math>

:<math> \sin(x) = x - {x^{3} \over 3!} + {x^{5} \over 5!} - \cdots = \sum_{n=0}^{\infty} \frac{(-1)^nx^{2n+1}}{(2n+1)!}</math>

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

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

:<math>\displaystyle \sin(-x) = -\sin(x)</math>

:<math>\displaystyle \cos(-x) = \cos(x)</math>

:<math>\displaystyle \frac{d}{dx} \sin(x) = \cos(x)</math>

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