Periodic Phenomena

Read the problem carefully to find the highest value M and the lowest value m the quantity ever reaches. The amplitude A tells you how far the function swings from its middle (midline) value: A = (M − m) / 2. The midline D (vertical shift) is the average: D = (M + m) / 2. For example, if a tide reaches 4.5 m at high tide and 0.3 m at low tide, then A = (4.5 − 0.3)/2 = 2.1 m and D = (4.5 + 0.3)/2 = 2.4 m.

The period T is the length of one complete cycle — one full repetition of the pattern. From T you compute the angular frequency (also called the B-value): B = 2π / T. If the tide completes one full cycle every 12.4 hours, then B = 2π / 12.4 ≈ 0.507 radians per hour. Larger B means the function cycles faster; smaller B means it cycles more slowly.

The phase shift C adjusts where the cycle starts horizontally. If you choose a cosine model, C is the x-value (time, angle, etc.) at which the function reaches its maximum. If you choose a sine model, C is the x-value at which the function is at the midline and rising. In the equation f(x) = A sin(B(x − C)) + D, a positive C shifts the graph to the right; a negative C shifts it to the left. Always match your shift to your data: if the maximum occurs at t = 2 h, a cosine model gives C = 2.

Both sin and cos work — it is a matter of convenience. Use cosine when you know the time of a maximum (or minimum); use sine when you know when the function crosses the midline going upward. Then substitute A, B, C, D into f(x) = A sin(B(x − C)) + D or f(x) = A cos(B(x − C)) + D. Always check: plug in the known maximum, minimum, and midline crossings to verify your equation produces the correct outputs.

Once the equation is written, use it to answer questions: What is the value at a specific time? When does the quantity first reach a given level? How many cycles occur in a given interval? Solve for x algebraically using inverse trigonometric functions, or read approximate answers from a graph. Always attach units and interpret answers in the context of the original phenomenon.

• A — Amplitude (same as f): Half the distance between the maximum and minimum values; always taken as positive. A = (max − min)/2.
• T — Period (time (s, h, days, etc.)): The length of one complete cycle. Determines how quickly the function repeats.
• C — Phase Shift (same as t): Horizontal shift; the value of t at which the cosine reaches its maximum (for a cosine model).
• D — Vertical Shift (Midline) (same as f): The average of the maximum and minimum; the horizontal axis around which the function oscillates. D = (max + min)/2.
• \frac{2\pi}{T} — Angular Frequency (B) (radians per unit time): Controls how many radians of the cycle pass per unit of time. Larger values compress the graph horizontally.

Problem. A Ferris wheel has a diameter of 40 m. The center of the wheel is 25 m above the ground. The wheel completes one full rotation every 80 seconds. A rider's seat is at the 3 o'clock position (the rightmost point, which is at the height of the center) when t = 0, and the wheel rotates counterclockwise (so the rider rises first). Write a sinusoidal function h(t) for the rider's height above the ground, then find the rider's height at t = 30 s.

Answer. The rider's height at t = 30 s is approximately 39.1 m above the ground.

• Amplitude and Period of Sinusoidal Functions — Amplitude and period are the two most important parameters you extract from a periodic phenomenon before writing its equation.
• Phase Shift and Vertical Shift — Phase shift and vertical shift complete the four-parameter model, positioning the sinusoidal curve to match real data.
• Graphs of Sine and Cosine Functions — Being able to sketch and read sinusoidal graphs is essential for interpreting periodic phenomena visually.
• Inverse Trigonometric Functions — Solving 'when does the quantity reach a given value?' requires applying inverse trig to the sinusoidal model equation.