Sinusoidal modelling with seasonal data (IB Math AI HL)

Key ideas

• a is half the distance from the maximum to the minimum; d is the middle value.
• Period = 2π ÷ b, so b = 2π ÷ period. For a yearly cycle in days, b = 2π ÷ 365.
• c is a horizontal shift: for a sine model, the time when the curve crosses the principal axis going up.
• Sine regression on the GDC needs radian mode for b in radians, and enough points to cover at least one full cycle.
• To find how long the model is above a level, solve the equation for the two crossing times in one cycle and subtract.

Worked example

In a city the hours of daylight on day t of the year are modelled by H(t) = 4.5 sin(2π(t − 81) ÷ 365) + 12. Write down the greatest and least daylight, find the day of greatest daylight, and find for how many days daylight is more than 14 hours.

1. Greatest = 12 + 4.5 = 16.5 hours; least = 12 − 4.5 = 7.5 hours.
2. The sine is 1 when 2π(t − 81) ÷ 365 = π/2, so t − 81 = 365/4 = 91.25 and t ≈ 172.
3. H(t) > 14 when sin(2π(t − 81) ÷ 365) > 4/9.
4. The angle is between 0.4606 and π − 0.4606 = 2.6810, so t is between about 107.8 and 236.7.
5. 236.7 − 107.8 ≈ 129 days (days 108 to 236).

Answer: 16.5 hours and 7.5 hours; greatest daylight on about day 172; about 129 days have more than 14 hours.

Exam-style HL AI questions

• HL AI Geometry and Trigonometry questions by topic — worked examples and practice questions for every topic in this unit

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 = 2 sin x + 1
• Sinusoidal regression
• Open the free Grapher

Maths IA ideas that use sinusoidal modelling with seasonal data

• Can a sum of sine waves predict the tides?
• Is crime seasonal? Testing monthly crime counts
• How does the shape of the daylight curve change with latitude?
• Modelling the sun's height from a stick's shadow
• All HL AI IA ideas

Related in Geometry and Trigonometry

• Graph theory — adjacency matrices, spanning trees and route problems
• Voronoi diagrams — construction and applications
• 3D vectors — dot product and cross product

Revise Sinusoidal modelling with seasonal data for the exam

• HL AI revision notes — every unit summarised, with the key results to learn
• HL AI past papers — specimen papers and timed practice papers
• Formulas for Geometry and Trigonometry — on the one-page HL AI formula sheet (free PDF), with the ones not in the IB booklet marked
• HL 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 S89 · Problem S24 · Problem S88 · Problem S23. All 22 →