Mathematics 11–12 · Years 11–12

Sun elevation from a shadow: inverse tan checked against Geoscience Australia

Trigonometry and measure of angles: Trigonometry with acute angles (Mathematics Advanced, Year 11); Inverse trigonometric functions: Definitions of inverse trigonometric functions (Mathematics Extension 1, Year 12: only the final step, the domain and range of tan^-1)

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The idea

The ratio of a stick's height to its shadow passes through the inverse tangent to give the sun's elevation, and Geoscience Australia's calculator gives the same angle from astronomy.

What you need

A 1.000 m stick held vertical with a plumb line or set square on level paving
A tape, chalk, a watch set to standard time
The Geoscience Australia Sun and Moon Azimuth and Elevation calculator

How to do it

On a sunny day mark the tip of the stick's shadow every 30 minutes from 9 am to 3 pm and record the time and the shadow length s.
Compute the elevation theta = tan^-1(1/s) for each reading and plot theta against time.
Read the time of the shortest shadow and the greatest elevation.
Enter the site, date and times in the Geoscience Australia calculator and compare its elevations with yours.
Explain why the inverse tangent, not the tangent or its reciprocal, is needed, and state its domain and range.

What you should see

The noon elevation is 90 - |phi - delta|, which in Sydney (latitude 33.87 degrees south) is 90 - 33.87 - delta, because the sun's declination delta never exceeds 23.44 degrees either way: 56.1 degrees at the equinoxes (a 0.671 m shadow from a 1 m stick), 79.6 degrees at the December solstice (0.184 m) and 32.7 degrees at the June solstice (1.558 m). At s = 0.671 m a 1 cm error in locating the fuzzy shadow tip changes the elevation by 0.4 degrees. The learner knows it worked when the plotted curve peaks near the time of greatest elevation in the calculator's values and the readings agree with the calculator to within that reading error.

What changes

What you change: time of day
What you measure: shadow length and sun elevation
What you keep the same: stick exactly vertical and 1.000 m
level paving
the same reading point on the shadow tip

Common misconceptions

Each of these ideas is wrong, and the activity is a chance to test it.

tan^-1(x) means 1/tan(x); the inverse function undoes tan, it is not its reciprocal.
The sun is directly overhead at noon in Sydney; its highest elevation there is 79.6 degrees.

Safety card

Low riskLearners carry it out

Hazards

sun exposure
looking towards the sun

Controls

hats and sunscreen
read the shadow only; never look at the sun

Note

No chemicals and no heat are used, so the NSW Department of Education Chemical Safety in Schools package does not apply; outdoor work follows the school's grounds or excursion risk assessment (RiskAssess where the school uses it).

Curriculum references

The NSW syllabus outcomes and Australian Curriculum v9 codes this activity supports. They are references, not a verified or complete curriculum alignment.

Mathematics Advanced 11–12 Syllabus (2024), Year 11 focus area Trigonometry and measure of angles; Year 11 taught from Term 1 2026, Year 12 from Term 4 2026, first HSC examination 2027 (the 2017 syllabus is still taught to Year 12 until then); page read 2026-09-22MAV-11-04
Mathematics Extension 1 11–12 Syllabus (2024), Year 12 focus area Inverse trigonometric functions; Year 11 taught from Term 1 2026, Year 12 from Term 4 2026, first HSC examination 2027 (the 2017 syllabus is still taught to Year 12 until then); page read 2026-09-22ME1-12-03
Australian Curriculum v9No Australian Curriculum v9 code is listed.

Sources

The pages the author read to write this activity.

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