IB Maths IA idea · Geometry, maps and space

Day length: how does latitude change the year?

Near the equator, day length barely changes all year. Near the Arctic Circle, it swings from almost nothing to almost everything. One equation explains both, and the tilt of the Earth is in it.

Geometry, maps and space
SL ideal
HL ideal
Maths AA Maths AI Published data
Syllabus: AA 2.10, AA 3.4, AA 3.5, AA 3.7, AA 3.8, AA 3.9, AA 5.6

By Photon Academy · Updated 13 November 2026

These ideas are starting points, not an IA to copy. Never copy an idea blindly: make the question, method and analysis your own, and check your plan with your teacher.

The big idea

The Earth's axis is tilted, so the Sun's angle above the equator, the declination, changes through the year in a smooth, sine-like way. Combine that with your latitude and a little spherical geometry and you can get the exact time between sunrise and sunset as a function of the day of the year. You can also ignore the physics and fit a sine curve to real sunrise tables. Which model is better, and where does each break? The formula is routine. The marks come from testing it: fit both models to real data, measure the residuals, explain the places they fail (polar day, twilight, atmospheric refraction) and say what you would change.

The maths you will use

Sinusoidal model of declination. Write the declination as $\delta = 23.44\sin\!\left(\frac{360(284+n)}{365}\right)$ degrees and explain each number.
Trigonometry on a cross-section. From the sunrise geometry, derive $\cos\omega_0 = -\tan\varphi\tan\delta$ and day length $2\omega_0/15$ hours.
Fitting a sine curve. Fit $L = a + b\sin(c(n - d))$ to real day lengths, with technology, and read off the amplitude and shift.
Residual analysis. Plot residuals of each model against the day of the year; compare the largest errors in minutes.
Rate of change. Differentiate the model to find minutes gained per day and the day it is largest (AA HL: with inverse trigonometry).

Key equations

\delta(n) = 23.44\sin\!\left(\frac{360(284+n)}{365}\right) \text{ degrees}

\cos\omega_0 = -\tan\varphi\,\tan\delta

L = \frac{2\omega_0}{15} \text{ hours}

The maths in action

Left: three curves of day length over a year, nearly flat for Singapore, a medium sine for London and a large sine for Reykjavik. Right: day length on 21 June rising with latitude to 24 hours at the polar circle, and the mirror image for 21 December. — (a) Modelled day length through the year: Singapore varies only from 11.92 to 12.08 hours, London from 7.6 to 16.4 and Reykjavik from 3.6 to 20.4. (b) Day length on the solstices against latitude; the Sun stays up or down all day beyond 66.56 degrees.

Example research question

To adapt, never to copy

For two places at different latitudes, how well does a day-length model built from the sunrise equation and a sinusoidal declination match a year of real sunrise and sunset times, and how does it compare with a sine curve fitted directly to the data?

Change the context, the data or the method so the question is yours. Our Maths IA guide shows how to check it.

Check your research question

Names the places and the year of data, and where the sunrise and sunset times come from.
Names the model or models you compare and what each parameter means.
Says how you will judge the fit: a residual measure, or the largest error in minutes.
Says what 'sunrise' means in your data and your model, since they may not match.
Is narrow enough for 12 to 20 pages and goes beyond fitting one sine curve once.

Which course it fits

AA SL ideal
AA HL ideal
AI SL ideal
AI HL ideal

Syllabus links in Maths AA:

AA 2.10 Solving equations graphically and analytically
AA 3.4 Radians, arc length and sector area
AA 3.5 Unit circle definitions of sine, cosine and tangent
AA 3.7 Circular functions and their graphs
AA 3.8 Solving trigonometric equations
AA 3.9 Reciprocal and inverse trigonometric functions (HL only)
AA 5.6 Standard derivatives; chain, product and quotient rules

Steps

Read and predict. Find out what declination is. Before any calculation, guess how the swing in day length changes as you move away from the equator.
Choose places and data. Pick two or three places at different latitudes, and download a full year of sunrise and sunset times from a source you can cite.
Clean the data. Convert times to hours, compute day length, and decide what to do with time zones and clock changes.
Trial run (1 lesson). Plot one year of day length for one place and fit a sine by hand. Fix any problem now: units, time zone, the start of the year.
Derive the physical model. Draw the cross-section, derive the day length from latitude and declination and check it on the equinox and the solstices.
Compare the models. Put the fitted sine and the physical model on one graph with the data, and measure the residuals of each.
Go further, with a reason. Pick one extension: the rate of change, polar day, or a refraction correction. Say why it adds to your question.
Reflect as you go. After every result, write one line on what it means for a real place and what it assumes. These lines become your evaluation.
Cut, check, then send. Check every graph has labelled axes with units, every symbol is defined once and the conclusion answers your aim. Then send your teacher one draft.

Make it yours

Derive from the geometry. Draw the Earth in cross-section and work out the length of the day from the latitude and the Sun's declination.
Fit a sine to real tables. Take a year of sunrise and sunset times for one place and fit a sine curve to day length by hand or with technology.
Two places, one model. Compare a place near the equator with one at a high latitude using the same formula.
Rate of change of day length. Find how many minutes per day the day length changes, and when that rate is largest.
Polar day and night. Find the latitude where the Sun stays up on the solstice, and how many days of midnight sun a place gets.
Include refraction. Replace 'the Sun's centre is on the horizon' by a more realistic sunrise definition and see how the day lengths change.

HL extension

Differentiate the day length to find the rate of change and show it is largest at the equinoxes, handle the polar cases with a piecewise model, and add a refraction correction to measure how much better it fits the data.

The data you need

A full year of sunrise and sunset times for two or three places from a public almanac or calculator, converted to day length in hours.

Traps

Fitting a sine curve with the wrong period, or leaving the shift out and getting the longest day in March.
Putting a calculator in degrees for one step and radians for the next.
Using clock time, with daylight saving changes, as if it were solar time.
Forgetting that the inverse cosine is undefined when the Sun never sets or never rises.
Calling a sine fit 'proved' because R-squared is high, without looking at the residuals.
Not saying what counts as sunrise in the data you use.

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