Complex Numbers
I originally wasn’t going to make a post about this topic, but I realized that trigonometry is so connected to complex numbers, that it is a natural follow-up.
Definitions of the Complex Numbers
The normal numbers that we use on a daily basis ($0$, $1$, $2$, $-1/12$, $\pi$, etc) are called real numbers. The set of all real numbers is denoted with the symbol $\mathbb{R}$. One property of squaring real numbers is that the result is always positive. There is seemingly no square root of a negative real number. Thus, mathematicians decided to define such a number into existence. The variable $i$ is defined as
\[i^2 + 1 = 0 \qquad\text{or}\qquad i^2 = -1\]$i$ is called an imaginary number. The set of all imaginary numbers is denoted with the symbol $\mathbb{I}$. Now, suppose we want to take the square root of $-64$, we can do that as follows
\[\sqrt{-64} \ = \ \sqrt{64 \cdot (-1)} \ = \ \sqrt{64} \cdot \sqrt{-1} \ = \ 8i\]Consider what happens when we combine real and imaginary numbers.
\[z = x + iy \qquad\text{For any } x, y \in \mathbb{R}\]Notice that there is no way to further simplify the right-hand side because real numbers and imaginary numbers do not interact via addition/subtraction. We call $z$ a complex number because it contains a real number component and an imaginary number component. The set of all real numbers is denoted with the symbol $\mathbb{C}$.
The Complex Plane
Since the real and imaginary components of complex numbers are independent of each other, we can think of complex numbers as points on the 2D plane. The x-axis is the real number line, and the y-axis is the imaginary number line. This is called the complex plane.
This should look very familiar. We could instead express this same point using the real 2D plane.
Therefore, we can instead express complex numbers as the following
\[z = r\cos \theta + i r \sin \theta = r(\cos \theta + i \sin \theta) \qquad\text{For } r, \theta \in \mathbb{R} \text{ and } r > 0\]Now, instead of thinking of complex numbers as one real component and one imaginary component, we can think of them as a magnitude and a direction in the complex plane. Thus, the above is often shorted as $z = r \ cis \ (\theta)$.
We now have two different, but equivalent, ways to write a complex number. $z = x + iy$ is call the Cartesian form and $z = r \ cis \ \theta$ is called polar form. It’s important to know how to translate between them because they are useful in different situations. This is just a simple application of trigonometry and the Pythagorean theorem.
\[\begin{align} &x = r \cos \theta &\qquad\qquad& r = \sqrt{x^2 + y^2} \\[10pt] &y = r \sin \theta &\qquad\qquad& \tan \theta = \frac{y}{x} \end{align}\]Addition, Multiplication, and Exponentiation of Complex Numbers
The addition of complex numbers occurs component-wise, thus we should use Cartesian form. If $z = x + i y$ and $w = a + i b$, then
\[z + w = (x + a) + i (y + b)\]Multiplication of complex numbers is where things get interesting. Here, it is best to use polar forms. If $z = r(\cos \theta + i \sin \theta)$ and $w = s (\cos \phi + i \sin \phi)$, then
\[\begin{align} z \cdot w &= r(\cos \theta + i \sin \theta) \cdot s (\cos \phi + i \sin \phi) \\[10pt] &= rs(\cos \theta \cos \phi + i \sin \theta \cos \phi + i \sin \phi \cos \theta + i^2 \sin \theta \sin \phi) \\[10pt] &= rs((\cos \theta \cos \phi - \sin \theta \sin \phi) + i (\sin \theta \cos \phi + \sin \phi \cos \theta)) \\[10pt] &= rs(\cos (\theta + \phi) + i \sin (\theta + \phi)) \\[10pt] \end{align}\]Thus, multiplying $z$ by $w$ rotated $z$ by the angle of $w$ and scaled $z$ by the magnitude of $w$.
Finally, we take a special case of the multiplication rule to get the power rule. If $z = r (\cos \theta + i \sin \theta)$ is a complex number, and $n$ is a positive integer, then
\[z^n = r^n (\cos(n \theta) + i \sin(n \theta))\]