Sinusoidal models for periodic data (IB Math AI SL)

Key ideas

• Amplitude: a = (max − min) ÷ 2. Principal axis: d = (max + min) ÷ 2.
• Period = 360° ÷ b (or 2π ÷ b in radians), so b = 360° ÷ period. The time from a maximum to the next minimum is half a period.
• c is a horizontal shift. A cosine model starts at a maximum at x = c, so it is often the easier choice when you are given the time of a maximum.
• Use the units of the question: if b came from 360°, the GDC must be in degrees.
• Check your model at one known point before using it to predict.

Worked example

The depth of water in a harbour is 9.2 m at high tide at 03:00, and 2.8 m at the next low tide at 09:15. Model the depth by D(t) = a cos(b(t − c)) + d, where t is hours after midnight, and find the depth at 12:00.

1. a = (9.2 − 2.8) ÷ 2 = 3.2 and d = (9.2 + 2.8) ÷ 2 = 6.
2. High to low tide takes 6.25 hours, half a period, so the period is 12.5 hours and b = 360 ÷ 12.5 = 28.8 (degrees per hour).
3. The maximum is at t = 3, so c = 3: D(t) = 3.2 cos(28.8(t − 3)) + 6.
4. Check: D(9.25) = 3.2 cos(180°) + 6 = 2.8 m.
5. D(12) = 3.2 cos(259.2°) + 6 = 5.40 m (3 s.f.).

Answer: D(t) = 3.2 cos(28.8(t − 3)) + 6, and the depth at 12:00 is 5.40 m.

Exam-style SL AI questions on this topic

• Sinusoidal Modelling questions — 13 questions with mark schemes, plus free worked examples
• All SL AI Functions questions by topic

Do it on your GDC

A GDC is allowed on every AI paper. Keystrokes for the TI-84 Plus CE, TI-Nspire CX II and Casio fx-CG50, with a checked worked example:

• Trig graphs and angle mode on your calculator
• Regression on your calculator

See it on a graph

• Graph of y = sin x
• Sinusoidal regression
• Open the free Grapher

Maths IA ideas that use sinusoidal models for periodic data

• Can a sum of sine waves predict the tides?
• Is crime seasonal? Testing monthly crime counts
• Which function best describes the arch of a famous bridge?
• How does the shape of the daylight curve change with latitude?
• All SL AI IA ideas

Related in Functions

• Piecewise functions and modelling
• Linear functions and modelling
• Quadratic functions — vertex, roots, and applications

Revise Sinusoidal models for periodic data for the exam

• SL AI revision notes — every unit summarised, with the key results to learn
• SL AI past papers — specimen papers and timed practice papers
• Formulas for Functions — on the one-page SL AI formula sheet (free PDF), with the ones not in the IB booklet marked
• SL AI formula booklet by topic — every formula in the IB booklet's notation, with a free PDF
• IB Maths prediction papers — full papers in the IB format, with markschemes

Stretch yourself: original algebra problems at Senior level (ages 16 to 18), with hints and full solutions: Problem S27 · Problem S91 · Problem S26 · Problem S90. All 22 →