⏱️ Trigonometry and Time Periodic Functions

Trigonometric functions model cyclical phenomena across nature and technology, providing powerful tools for analyzing time-based patterns.

Fundamental Model

f(t) = A·sin(2πft + φ) + C
Where:
A = amplitude
f = frequency (cycles/time)
φ = phase shift
C = vertical offset

Key Applications

FieldApplication
Electrical EngineeringAC current (60Hz sine waves)
AstronomyTidal height predictions
BiologyCircadian rhythm analysis
EconomicsSeasonal demand forecasting

Tide Prediction

Modeling daily tides at Seattle:
h(t) = 2.5sin(πt/6 + 4π/3) + 4.2
Where t is hours from midnight

Fourier Analysis

Decomposing complex waveforms into sine components:

Any periodic function = Σ(Aₙsin(nωt) + Bₙcos(nωt))

Real-World Fact: The human ear performs a biological Fourier transform, with different cochlear regions responding to specific frequencies.

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