NapkinCalc

Trigonometry B — Waves & the Triangle Laws

Sine waves

ELI5: walk around the circle at a steady pace and plot your height over time — you get a sine wave. Two knobs shape it:

  • amplitude stretches it tall
  • angular frequency ω squeezes the repeats closer; one full cycle takes a period of 2π/ω.
amp=2amp = 2 amplitude — try 0.5 or 4
ω=1.5\omega = 1.5 angular frequency — try 3
amp * sin(omega * x)
-101-505

y = amp · sin(ω·x) — taller with amp, tighter with omega

Real-world hook: sine waves are sound (frequency = pitch, amplitude = loudness), AC mains electricity, light, ocean swells, and the daylight hours over a year.

Try it yourself: for sin(ω·x) with ω = 4, how long is one full period (2π/ω)?

periodyou=π2period_{you} = \frac{\pi}{2} = 1.57081.5708 ✏️ Your turn: find the period of sin(4x). The check verifies period × 4 = 2π — so you must compute 2π/4 yourself.
✓ pass abs(periodyou⋅4−2⋅π)<0.000001\mathrm{abs}\left(period_{you} \cdot 4 - 2 \cdot \pi\right) < 0.000001 green when period × ω = 2π