ELI5: complex trigonometry

What is complex trigonometry?

Complex trigonometry is like regular trigonometry (which uses triangles), but instead of just using regular numbers, we use special numbers that have both a regular part and an "imaginary" part [1]. Imagine a game where you can move forward-backward AND sideways at the same time—that's what complex numbers let you do [3].

Why do we need it?

Regular trigonometry only works with normal numbers, but scientists and engineers discovered that many real-world problems (like waves and spinning things) are easier to solve if we allow these special numbers [4]. It's like having a superpower tool that makes hard problems simpler [1].

How does it connect to angles?

Complex numbers can represent angles and rotations on a special diagram called an Argand diagram, where one direction is "regular" and another direction is "imaginary" [5]. When you rotate or spin something using complex numbers, the math becomes much cleaner than using regular trigonometry [3].

What's the magic formula?

Euler's formula is the secret sauce—it shows that trigonometric functions (like sine and cosine) are actually the same as complex exponentials, which means rotating and scaling in the complex plane [4][3].