I am preparing a post and accompanying code to demonstrate the usefulness of using complex analysis in the context of computer programming. In order to prepare the blog post, I had to re-learn some trigonometry. Here’s the main facts I needed as a refresher.

## Complex Number

A complex number $z$ is a number with “real” $Re(z)$ and “imaginary” $Im(z)$ parts. A complex numbers allows us to perform algebra with the $\sqrt{-1} = i$ as a well-defined concept.

Since complex numbers are simple to think about as rotations on the Cartesian XY plane, it’s useful to recall the basic trigonometric functions and their common formulae.

## Unit Circle

Draw a right triangle onto the complex plane with $a = b$ and hypotenuse $r = 1$. The Pythagorean Theorem gives us lengths for $a$ and $b$ as

Division by an irrational number is a computationally fraught exercise. Try it by hand and think about what’s involved: an infinite sequence of divisions *plus one more* once you’re done. So it’s easier to multiply by identity to bring the irrational number up top, like so

from which point you can find a suitable approximation for $\sqrt{2}$ and divide that approximation by $2$.

Note that there are two solutions $\frac{\sqrt{2}}{2}$ and $-\frac{\sqrt{2}}{2}$. We’ll just take the positve solution for now (the negative is simply reflected across the origin).

## Basic Functions

From the above, we have a right triangle inscribed in the unit circle with sides of length $a = \frac{\sqrt{2}}{2}$, $b = \frac{\sqrt{2}}{2}$, and $r = 1$. And since $a = b$, the angles of the triangle must be $\pi / 4$, $\pi / 4$, and $\pi / 2$ adding up to $\pi$ or 180 degrees.

All this information allows us to describe the trigonometric functions

These functions have “flipped” versions

## Inverse Functions

Inverse trig functions are described by the definitions

For instance, if you know that $\sin \theta = \frac{\sqrt{2}}{2}$ then you can find $\sin^{-1} \frac{\sqrt{2}}{2} = \frac{\pi}{4}$. Why? Because $\sin \frac{\pi}{4} = \frac{\sqrt{2}}{2}$ as computed previously.

Trigonometic tables are usually employed rather than re-compute the functions and their inverese repeatedly; understanding the relation, however, is still important when simplyfying expressions.

## Summation and Difference Formulae

Not that the symbol $\mp$ is *not* a typo; the sign switches in those cases, eg. $\cos(\alpha + \beta) = \cos \alpha \cos \beta - \sin \alpha \sin \beta$.

It’s best to commit these to sum/difference formulae to memory or keep handy someplace.

## More Trigonometry

Here’s a useful cheat sheet to refresh your memory: Trig Cheat Sheet by Paul Dawkins.