The idea, step by step
Test geometric conditions independently
Triangle angles are positive and sum to 180°. The longest side faces the largest angle, and the sum of the two shorter lengths exceeds the longest. Check these conditions even when a calculator returns a number.
For acute-angle SSA data, b sinA is the perpendicular height relative to the other known side. If the opposite side a is below that height, it cannot reach the baseline; other comparisons determine one or two candidates.
A result must answer the stated problem
Retain intermediate precision and round once at the end. A measured angle already has uncertainty; extra calculator digits do not improve the measurements.
A sine or cosine outside its permitted range is not repaired by forcing it to ±1. Recheck the original measurements and the chosen formula. Explain any rejected candidate and include units and assumptions.
A common trap
Rounding sinB=1.05 down to 1 to obtain an angle.
Instead: If the value genuinely exceeds 1, there is no matching angle; recheck the model and data.
See the thinking
Worked example 1
A=40°, a=12 cm and b=7 cm. Decide how many triangles fit.
- sinB=7sin40°/12.
- The acute candidate is 22.0218°; the supplementary one is 157.9782°.
- The supplementary angle plus A exceeds 180°, so reject it. The acute candidate fits.
Worked example 2
A student claims a triangle has sides 5 m, 7 m and 13 m. Explain why no angle calculation is needed.
- The shorter sides add to 12 m.
- 12<13, so they cannot close a triangle around the longest side.
- Triangle inequality rejects the claim immediately.
Check your understanding
Side a=14 faces A=35°. Side b=9 faces B. Is B smaller or larger than A?
Show answer
Smaller.
The shorter side must face the smaller angle.
Your turn
6 questions · 16 marks · Work at your own pace.
Read the questions online
- Can lengths 7, 9 and 17 form a triangle? Explain.2 marks
- A=30°, a=10 and b=14. How many triangles fit, and what are the candidate angles B?3 marks
- A=50°, a=8 and b=13. Explain the absence of a real inverse-sine result.3 marks
- A triangle has A=80° and B=40°. Which of a and b must be longer?2 marks
- A length calculation gives 18.76491 m. Round to one decimal place and explain whether to round each intermediate value too.3 marks
- A proposed solution gives angles 60°, 70° and 55°. State a check that rejects it.3 marks
Read the worked solutions
Equivalent correct methods earn the corresponding marks. Include units where asked.
- Can lengths 7, 9 and 17 form a triangle? Explain.2 marks
7+9=16<17, violating triangle inequality.
No. - A=30°, a=10 and b=14. How many triangles fit, and what are the candidate angles B?3 marks
sinB=0.7.
Both candidates leave a positive third angle.
Two; B≈44.4° or 135.6°. - A=50°, a=8 and b=13. Explain the absence of a real inverse-sine result.3 marks
sinB=13sin50°/8≈1.245>1.
This is impossible geometry, not insufficient calculator precision.
The data are inconsistent; no triangle exists. - A triangle has A=80° and B=40°. Which of a and b must be longer?2 marks
Larger angle A faces longer side a.
a - A length calculation gives 18.76491 m. Round to one decimal place and explain whether to round each intermediate value too.3 marks
The hundredths digit is 6.
Early rounding may change the final result.
18.8 m; keep intermediate precision. - A proposed solution gives angles 60°, 70° and 55°. State a check that rejects it.3 marks
60+70+55=185.
Recalculate before accepting its side lengths.
Their sum is 185°, not 180°.
Curriculum reference & lesson scope
SACE Board of South Australia
Stage 1 General Mathematics: practical trigonometry, including similarity, right triangles, triangle area, sine/cosine rules and contextual interpretation.
Official curriculum source ↗