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.


  1. Trigonometry and Complex Exponentials, William Stein.