Trigonometry: The Math of Circles and Waves
If you have ever seen a character animate in a circle, heard a synthesized musical note, or watched a radar sweep across a screen, you have used trigonometry. The difference is that the computer knew it was using trigonometry, and you did not.
This guide fixes that. We are not going to drill triangle diagrams until you can recite the ratios. We are going to start from something you already understand - a point moving around a circle - and build up to the functions that describe every repeating pattern in nature and technology. By the end, sine and cosine will look like the natural language of anything that cycles.
This is the tenth guide in the Mathematics track. It assumes the coordinate geometry from Numbers & Number Systems and the function idea from Sets, Relations, and Functions. If you can plot a point on a graph and read a function, you are ready.
How to read this
- Here for the "what is this even for" answer? Start with Phase 1 - the unit circle and why these functions exist.
- Want the full toolkit? Read in order - waves build on the circle, and applications build on waves.
The phases
- The Unit Circle and Why Sine/Cosine Exist - a point moving around a circle, the birth of sine and cosine, radians instead of degrees, and the code that draws a circle.
- Waves, Frequencies, and the Real World - sine waves as the shape of every repeating phenomenon, amplitude and frequency, phase shifts, and generating audio tones with code.
- Rotation, Navigation, and Where Am I Facing - rotating 2D sprites, simple GPS triangulation, and the builder's guide to seeing trigonometry in graphics, audio, and game development.
This builds on Numbers & Number Systems (coordinates, angles) and Linear Algebra (rotation matrices). It is the geometry behind most visual and audio software.