Tides, sound waves, and circular motion all follow a sinusoidal (trig) function pattern. The standard form is:
\[ f(x) = A \cdot \sin(B(x - C)) + D \quad \text{or} \quad f(x) = A \cdot \cos(B(x - C)) + D \]
Each letter controls something different:
- A = Amplitude β half the distance between max and min
\(A = \dfrac{\text{max} - \text{min}}{2}\)
- B = frequency factor β related to period
\(B = \dfrac{2\pi}{\text{period}}\)
- C = Phase shift β horizontal shift left/right
- D = Midline (vertical shift) β average of max and min
\(D = \dfrac{\text{max} + \text{min}}{2}\)
π Example: Tide Problem (like the worksheet!)
Given: High tide = 24 ft, Low tide = 12 ft, Period = 11 hrs 5 min
- Convert period: 11 Γ 60 + 5 = 665 minutes
- Amplitude: \(A = \dfrac{24 - 12}{2} = \mathbf{6}\)
- \(B = \dfrac{2\pi}{665}\)
- Midline: \(D = \dfrac{24 + 12}{2} = \mathbf{18}\)
- No phase shift (starts at high tide at midnight): \(C = 0\)
- Result: \(f(x) = 6\cos\!\left(\dfrac{2\pi}{665}x\right) + 18\)