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Relations between cosine, sine and exponential functions

$\displaystyle e^{\pm i \theta}$ $\textstyle =$ $\displaystyle \cos(\theta) \pm i \sin(\theta)$ (45)
$\displaystyle \cos(\theta)$ $\textstyle =$ $\displaystyle \frac{1}{2} \left(e^{+i \theta} + e^{-i
\theta}\right)$ (46)
$\displaystyle \sin(\theta)$ $\textstyle =$ $\displaystyle \frac{1}{2i} \left(e^{+i \theta} - e^{-i
\theta}\right)$ (47)

From these relations and the properties of exponential multiplication you can painlessly prove all sorts of trigonometric identities that were immensely painful to prove back in high school

Robert G. Brown 2004-04-12