November 6, 2017

The basics

Just like everyone else on twitter, when I saw this diagram below my reaction was: “why didn’t they show this to me 25 years ago?” The lengths of the triangle legs correspond to the values of the trigonometric functions. Drag the point to change the diagram around.

Complex Exponentials FTW

We start from the formula for complex exponentials:

\[\begin{eqnarray*}e^{ix} & & = & \ \cos x + i \sin x \\ e^{-ix} & = \ \cos -x + i \sin -x & = & \cos x - i \sin x \end{eqnarray*}\]

We re-arrange this a bit to find a formula for cosines and sines in terms of the exponentials. This is maybe the two trigonometry formulas worth memorizing:

\[\begin{eqnarray*} \cos x & = & \frac{e^{ix}+e^{-ix}}{2} \\ \sin x & = & \frac{e^{ix}-e^{-ix}}{2} \end{eqnarray*}\]

Never memorize a formula for sines and cosines of sums again. Stein has some examples of how you can use this.