# Trigonometry

## The Basics

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

## Complex exponentials FTW

(This is my favorite trig trick.) Never memorize a formula for sines of sums again. Start from

$e^{ix} = \cos x + i \sin x$
$e^{-ix} = \cos -x + i \sin -x = \cos x - i \sin x$

From these two, you get that

$\cos x = \frac{e^{ix} + e^{-ix}}{2}$
$\sin x = \frac{e^{ix} - e^{-ix}}{2i}$

Now expressions like $\sin (a+b)$ are obvious to work out instead of big and scary. You only need memorize those formulas above, and from them you can derive many of the annoying high school formulas.

## References

1. Trigonometry and Complex Exponentials, William Stein.