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:

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

Trig Identities

From Euler's Formula one can get many identities such as:

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

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

Trig Functions in Exponential Form

Sometimes it is easier to reduce equations from exponentials to trig functions. This can be done simply by adding and subtracting Euler's formula with its conjugate. For cosine this looks like:

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

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

And similarly for sine:

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

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

And therefore tangent:

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