Unit 3: Trigonometric & Polar Functions

Sinusoidal Modeling

Fitting a sinusoidal function to periodic context and data.

Advanced25 min lesson4 min readUpdated August 12, 2026Author not yet attributed

Fitting a Wave to a Repeating Pattern

Exponential and logistic models both describe quantities that settle toward a fixed value. Some quantities never settle — they rise and fall on a predictable, repeating cycle instead. Before reading on: what four numbers would you need to pin down, to build a sine or cosine model of a repeating pattern from just its highest and lowest values and how often it repeats?

Definition — Sinusoidal Model

A sinusoidal model has the form y = A·cos(B(x − C)) + D, where amplitude A = (max − min)/2 is the distance from the midline to a peak, midline D = (max + min)/2 is the center of oscillation, period = 2π/B is the length of one full cycle, and C is the horizontal shift locating a maximum.

y = A\cos\big(B(x - C)\big) + D

Worked Example — Modeling Periodic Temperature Data

A location's average monthly temperature reaches a maximum of 82°F in month 7 and a minimum of 42°F in month 1, repeating every 12 months. Amplitude: A = (82 − 42)/2 = 20. Midline: D = (82 + 42)/2 = 62. Period 12 gives B = 2π/12 = π/6. Since a cosine model peaks exactly at x = C, set C = 7: y = 20cos((π/6)(x − 7)) + 62. Check the minimum, x = 1: (π/6)(1 − 7) = (π/6)(−6) = −π, and cos(−π) = −1, giving y = 20(−1) + 62 = 42 — matching. Check the maximum, x = 7: (π/6)(0) = 0, cos(0) = 1, giving y = 20 + 62 = 82 — matching.

Worked Example — Choosing Sine, Cosine, or a Negative, Based on the Starting Value

A Ferris wheel's height starts at its lowest point, 3 ft, at t = 0, reaches a maximum of 43 ft, and completes one revolution every 8 minutes. Amplitude: (43 − 3)/2 = 20. Midline: (43 + 3)/2 = 23. Period 8 gives B = 2π/8 = π/4. Since height starts at a minimum rather than a maximum, use −cos instead of cos: h(t) = −20cos((π/4)t) + 23 — check t = 0: −20cos(0) + 23 = −20 + 23 = 3, matching the minimum. Check t = 4, a half-period later, which should be the maximum: (π/4)(4) = π, cos(π) = −1, so h(4) = −20(−1) + 23 = 43, matching.

Graph Visualizer

Trace modePoint mode (click graph to plot)

Show x^2

Color

Expression

Domain & range

x min

x max

y min

y max

Precision (decimal places)2

Evaluate a point

Evaluate at x =

x^2 = 0

Tip

Read a model's starting condition before choosing its form: cos starts at a maximum, −cos starts at a minimum, sin starts at the midline heading up, and −sin starts at the midline heading down — matching the starting value to one of these four saves guessing a phase shift by trial and error.

Common Mistakes

Computing period directly as B, instead of solving B = 2π/period.
B is not the period — it's 2π divided by the period. A period of 8 gives B = 2π/8 = π/4, a much smaller number than 8 itself.
Setting C equal to whatever x-value is mentioned first in the problem, regardless of whether that's a maximum.
In y = A·cos(B(x − C)) + D, C must specifically be an x-value where the function reaches its maximum — using a different reference point requires switching to sin, −sin, or −cos instead, or solving for a genuine phase shift.

Key Takeaways

Amplitude A = (max − min)/2 and midline D = (max + min)/2 come directly from a sinusoid's highest and lowest values.
Period determines B through B = 2π/period; a shorter period means a larger B and a faster-repeating wave.
Whether the pattern starts at a maximum, a minimum, or the midline determines which of cos, −cos, sin, or −sin fits without an extra phase-shift calculation.

Summary

A sinusoidal model turns a repeating pattern into an equation. The next lesson works in the other direction — starting from a trigonometric equation and finding every input that satisfies it.

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