Mathfolis

Sinusoidal Function Context and Data Modeling

Unit 3 · Trigonometric and Polar Functions

What AP Precalc asks here

Sinusoidal modeling assembles A, B, h, and k from the context: amplitude from half the vertical range, period from the cycle length, midline from the average of max and min, and phase shift from the starting position. Once the model is built, it predicts the quantity (height, temperature) at any time, and inverse trig recovers the time of any specified value.

Model recipe

Standard form
f(t)=Asin⁡(B(t−h))+kf(t) = A \sin(B(t - h)) + k
B from period
B=2πPB = \frac{2\pi}{P}
A, k from max/min
A=max⁡−min⁡2,    k=max⁡+min⁡2A = \frac{\max - \min}{2}, \;\; k = \frac{\max + \min}{2}
AP Tip: Pick sine for a model that starts at the midline going up; pick cosine for a model that starts at the maximum. Both forms are equivalent — just a different phase shift.
Type 1

Build a model from a context

Identify max, min, and period in the wording. Compute A, k, B; pick sin or cos based on starting position.

Example 1
A tide ranges from 22 ft (low) to 88 ft (high) with period 1212 hours. At t=0t = 0 the tide is at its midline rising. Which model fits? (A) h(t)=3sin⁡ ⁣(π6t)+5h(t) = 3 \sin\!\left(\tfrac{\pi}{6} t\right) + 5 (B) h(t)=3cos⁡ ⁣(π6t)+5h(t) = 3 \cos\!\left(\tfrac{\pi}{6} t\right) + 5 (C) h(t)=6sin⁡ ⁣(π12t)+4h(t) = 6 \sin\!\left(\tfrac{\pi}{12} t\right) + 4 (D) h(t)=3sin⁡(12t)+5h(t) = 3 \sin(12 t) + 5

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Type 2

Evaluate at a time

Substitute tt and use special-angle values where possible.

Example 2
For h(t)=3sin⁡ ⁣(π6t)+5h(t) = 3 \sin\!\left(\tfrac{\pi}{6} t\right) + 5, find h(3)h(3). (A) 2 (B) 5 (C) 8 (D) 11

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Type 3

Solve for time

Set the model equal to the target. Solve for the sin/cos value, then for tt using inverse trig.

Example 3
For h(t)=3sin⁡ ⁣(π6t)+5h(t) = 3 \sin\!\left(\tfrac{\pi}{6} t\right) + 5, at what smallest positive tt is h(t)=6.5h(t) = 6.5? (A) t=1t = 1 (B) t=2t = 2 (C) t=3t = 3 (D) t=4t = 4

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Sinusoidal Function Context and Data Modeling | AP Precalculus — Mathfolis