Mathematics 11–12 · Years 11–12
Distance to an inaccessible point by the sine rule
Trigonometry (Mathematics Standard 2, Year 12); Trigonometry and measure of angles: Trigonometry with angles of any magnitude (Mathematics Advanced, Year 11)
School laboratory, not for home
In a school laboratory, with a teacher supervising, under the school's risk assessment. Not for home.
This site has no interactive model of its own. Where a step or a material names a Concept Studio model, simulation or tool, it has not been built; an external simulation a step names (for example PhET) is not part of this site.
The idea
Two angles measured from the ends of a known baseline fix a triangle, and the sine rule turns that triangle into the width of a river or oval the learner never crosses.
Safety card
Setting: In a school laboratory, with a teacher supervising, under the school's risk assessment. Not for home.
Hazards
- working near a road or water
Controls
- a teacher or other adult supervises the group throughout
- stay on the near side behind any fence line
- wear high-visibility vests near roads
Note
No chemicals and no heat are used, so the NSW Department of Education Chemical Safety in Schools package does not apply; fieldwork near water or a road is run under the school excursion risk assessment (RiskAssess where the school uses it).
What you need
- A 50 m tape, two ranging poles or cones, a sighting compass or phone compass read to 1 degree
- A tree or post across the oval, a creek or a fenced road as the inaccessible point C
How to do it
- Lay out a 50.0 m baseline AB on the near side with a pole at each end.
- From A read the bearings to B and to C; their difference is angle A. From B read the bearings to A and to C; their difference is angle B.
- Compute angle C = 180 - A - B, then AC = AB sin B / sin C and BC = AB sin A / sin C.
- Compute the perpendicular width from C to the baseline as AC sin A, and the triangle's area as (1/2) AB AC sin A.
- Where it is safe, tape the distance directly; otherwise repeat with a different baseline and compare.
What you should see
AB = 50.0 m, A = 62 degrees and B = 75 degrees give C = 43 degrees, AC = 70.8 m, BC = 64.7 m, a perpendicular width of 62.5 m and an area of 1563 m^2. With both angles 1 degree out in the same direction the width ranges from 59.4 m to 65.9 m, so a 1 degree compass fixes it to about plus or minus 3 m. The learner knows it worked when two different baselines agree within that tolerance.
What changes
- What you change
- baseline length and the two measured angles
- What you measure
- computed distances AC, BC and the width
- What you keep the same
- straight baseline
- the same far point C
- compass held level and away from steel
Common misconceptions
Each of these ideas is wrong, and the activity is a chance to test it.
- The sine rule needs a right angle; it works in any triangle.
- A short baseline is as good as a long one; a short baseline makes angle C small and the answer very sensitive to angle errors.
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 Standard 11–12 Syllabus (2024), Year 12 Standard 2 focus area Trigonometry; 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 2026-09-22MST-12-S2-04
- 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
- Australian Curriculum v9No Australian Curriculum v9 code is listed.
Sources
The pages the author read to write this activity.