Math Notes

Newton had to develop calculus to explain why apples fall from trees. Fortunately most of the mathematics needed to solve our problems are fully developed subjects. So please share your knowledge.

Euler's Formula

This is something very useful as it can be used to derive many functions and identities. The formula is:

$e^{i\theta }=cos\left(\theta \right)+isin(\theta )$

From this one can get many identities such as:

$|e^{i\theta }|=|cos\left(\theta \right)+isin(\theta )|$

$1=cos^{2}\left(\theta \right)+sin^{2}(\theta )$

Also one finds that:

$e^{i\theta }+e^{-i\theta }=cos\left(\theta \right)+isin(\theta )+cos(\theta )-isin(\theta )=2cos(\theta )$

$cos\left(\theta \right)={e^{i\theta }+e^{-i\theta } \over 2}$

And similarly:

$e^{i\theta }-e^{-i\theta }=cos\left(\theta \right)+isin(\theta )-cos(\theta )+isin(\theta )=2isin(\theta )$

$sin\left(\theta \right)={e^{i\theta }+e^{-i\theta } \over 2i}$