Why go further than sin and cos of acute angles?
Once sine and cosine are coordinates on the unit circle, the same functions model anything that repeats: sound pressure, seasonal daylight, alternating current, a spinning wheel’s height. Engineers tune amplitude (how tall the wave is) and period (how long one cycle lasts). The model
y = A sin(Bx)
puts those knobs in the formula: |A| is amplitude; period is 2π/|B|.
Real signals
- Audio — louder is larger amplitude; pitch relates to frequency (linked to B).
- Day length over a year — roughly wave-like around a midline.
- Ferris wheel height vs time — sinusoidal in the ideal model.
- Heart monitors and sensors — repeating patterns analyzed with wave language.
Wave controls
Interactive
A and B
Try it: Raise A for taller waves. Raise B to fit more cycles in the window (shorter period).
amplitude |A| = 1
period 2π/|B| ≈ 6.28
Identities as rewrite rules
The Pythagorean identity sin²θ + cos²θ = 1 is the unit circle in algebra form. Double-angle formulas let you rewrite sin(2θ) or cos(2θ) when a model or equation needs it. Treat them as tools—like factoring—not as decoration.
Practice — Honors → Advanced Trig
Open Hyper-Scale Lab