The Rotating Point

Pre-Calculus · Axiom Academy

Send one point walking around a circle, and out of its shadow fall the two functions trigonometry is built on: cosine and sine. One point, walking in a circle Almost every wave you can name — sound, light, a vibrating string, the swing of a pendulum, alternating current — traces back to one tiny idea you can watch happen: a single point traveling around a circle. Trigonometry is the language for that motion, and the whole subject starts with one circle of radius 1. Watch the point ride once around the unit circle. At every moment it casts two shadows — straight down onto the horizontal axis, and straight across onto the vertical axis. Keep your eye on where those shadows land. That horizontal shadow is the number we call cos θ ; the vertical shadow is sin θ . The point's address on the circle is exactly (cos θ, sin θ) — the two shadows ARE the coordinates. Now you steer. Drag the angle dial — or grab the point and slide it around the rim. The point never leaves the unit circle, so reading its two coordinates is reading cos θ and sin θ directly. Stop at the marked angles and watch the numbers land on values worth remembering. The point's coordinates obey cos² θ + sin² θ = 1 at every angle — that's just the circle having radius 1.

This is the written version of the interactive lesson above. See the full Pre-Calculus course.