# 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

- Trigonometry and Complex Exponentials, William Stein.