Radians & Sine Graphs
Degrees are for protractors. Radians measure an angle by how far you've walked around the circle — and once you use them, the sine graph and all its transformations snap into place. For senior high school.
A radian is a distance
Walk around a circle of radius 1. The distance you've travelled along the rim is your angle in radians. A full lap is the full circumference, 2π ≈ 6.28 — so half a lap is π, a quarter lap is π/2, and the protractor's 60° becomes π/3.
Spin the circle, shape the wave 🟢
The three dials on a sine wave
A — amplitude, B — period
A is the circle's radius: taller circle, taller wave. B spins the circle faster, so the wave repeats every 2π/B — that's the period. Set B = 2 and watch two full waves fit where one used to.
C — phase shift
C slides the whole wave sideways — the circle simply starts its lap late (or early). Like every "inside the brackets" number, it works backwards: x − π/6 moves the wave right by π/6.
These are the same four dials as the Function Transformer — sin x is just another parent function. What radians buy you: the period formula 2π/B only works because angle and distance are the same thing.