Sinusoidal Modeling

In Graphs of Sine and Cosine, you learned the general form $y = A\sin(B(x - C)) + D$ and what each parameter controls. This page teaches you to go the other direction: given real-world data that repeats in a pattern, find $A$, $B$, $C$, and $D$ to create a mathematical model.

The Four Parameters from Data

Every sinusoidal model requires four values. Here is how to extract each one from data:

Worked Example 1: Daylight Hours

In a certain city, June 21 (day 172) has the most daylight at 15.2 hours, and December 21 (day 355) has the least at 9.0 hours. Model the daylight as a function of day number.

Step 1 — Find $D$ (midline):

$D = \frac{15.2 + 9.0}{2} = \frac{24.2}{2} = 12.1 \text{ hours}$

Step 2 — Find $A$ (amplitude):

$A = \frac{15.2 - 9.0}{2} = \frac{6.2}{2} = 3.1 \text{ hours}$

Step 3 — Find $B$ (from the period). The cycle repeats every 365 days.

$B = \frac{2\pi}{365} \approx 0.01721$

Step 4 — Find $C$ (phase shift). The maximum occurs on day 172. For a cosine function, the peak is at $x = C$, so:

$C = 172$

Using cosine (since we identified the maximum first):

$\boxed{y = 3.1\cos\!\left(\frac{2\pi}{365}(x - 172)\right) + 12.1}$

Verify: At $x = 172$ (June 21): $y = 3.1\cos(0) + 12.1 = 3.1 + 12.1 = 15.2$ ✓

At $x = 355$ (Dec 21): $y = 3.1\cos\!\left(\frac{2\pi}{365}(183)\right) + 12.1 \approx 3.1\cos(\pi) + 12.1 = -3.1 + 12.1 = 9.0$ ✓

Worked Example 2: Tide Height

A coastal harbor records the following tide data:

A dock worker needs to know: when is the tide at least 6 feet?

Step 1 — Identify max and min. Max $= 7.8$ ft (at $t = 3$), min $= 0.6$ ft (at $t = 9$).

Step 2 — Find $D$:

$D = \frac{7.8 + 0.6}{2} = 4.2 \text{ ft}$

Step 3 — Find $A$:

$A = \frac{7.8 - 0.6}{2} = 3.6 \text{ ft}$

Step 4 — Find the period. From max at $t = 3$ to the next time the pattern would reach the same max: the data shows a half-period from $t = 3$ (max) to $t = 9$ (min) is 6 hours. So the full period is 12 hours.

$B = \frac{2\pi}{12} = \frac{\pi}{6}$

Step 5 — Find $C$. The maximum is at $t = 3$. Using cosine:

$C = 3$

Model:

$h(t) = 3.6\cos\!\left(\frac{\pi}{6}(t - 3)\right) + 4.2$

Solving the dock worker’s question: When is $h(t) \ge 6$?

$3.6\cos\!\left(\frac{\pi}{6}(t - 3)\right) + 4.2 \ge 6$

$\cos\!\left(\frac{\pi}{6}(t - 3)\right) \ge \frac{1.8}{3.6} = 0.5$

Cosine is $\ge 0.5$ when the angle is in $[-\frac{\pi}{3}, \frac{\pi}{3}]$ (within one period).

$-\frac{\pi}{3} \le \frac{\pi}{6}(t - 3) \le \frac{\pi}{3}$

$-2 \le t - 3 \le 2$

$1 \le t \le 5$

Answer: The tide is at least 6 feet from 1:00 AM to 5:00 AM and again from 1:00 PM to 5:00 PM (each 12-hour cycle).

Choosing Sine vs. Cosine

Both model the same wave — the only difference is the default starting position:

• Cosine starts at a maximum: $\cos(0) = 1$. Use cosine when you can easily identify where the max occurs.
• Sine starts at the midline going up: $\sin(0) = 0$. Use sine when you can identify where the data crosses the midline while increasing.

Either function works for any data set — one just gives a simpler phase shift $C$ than the other.

Checking Your Model

After building a model, always plug in at least two known data points to verify:

1. Check the maximum: does $y$ equal the max value at the right $x$?
2. Check the minimum: does $y$ equal the min value at the right $x$?
3. Check a midline crossing: does $y$ equal $D$ at a point between max and min?

If any check fails, the most common errors are:

• Phase shift sign error (positive $C$ shifts right)
• Period error (using the half-period instead of the full period)
• Mixing up $A$ and $D$

Common Mistakes

1. Using the wrong period. The time from max to min is a half-period, not a full period. The full period is twice that distance.
2. Sign errors in the phase shift. In $y = A\sin(B(x - C)) + D$, the $C$ is subtracted. A positive $C$ shifts the graph right. Students often get the direction backwards.
3. Forgetting that $B$ multiplies the entire $(x - C)$. The phase shift is $C$, not $C/B$. If you write $y = A\sin(Bx - C) + D$ instead of $y = A\sin(B(x - C)) + D$, the shift is $C/B$, not $C$.
4. Not converting time units consistently. If the period is in hours, make sure all inputs are in hours. Mixing hours and minutes (or days and months) breaks the model.

Practice Problems

Test your understanding with these problems. Click to reveal each answer.

Key Takeaways

• To build a sinusoidal model: find $D$ (midline), $A$ (amplitude), $B$ (from the period), and $C$ (phase shift from where the max or midline crossing occurs)
• Use cosine when you identify the maximum first; use sine when you identify a midline crossing
• The time from maximum to minimum is a half-period — double it for the full period
• Always verify the model by plugging in at least two known data points
• Sinusoidal models apply to any repeating phenomenon: daylight hours, tides, temperature cycles, Ferris wheels, and AC voltage

Return to Trigonometry for more topics in this section.