Use the sine rule with correctly matched opposite pairs
Match each side with its opposite angle · Complete a triangle from two angles and one side · Reject angle sums that cannot form a triangle
Exam scope and assessment notes
Extended content; not a Core requirement. Historical study of Cambridge IGCSE Mathematics (US) 0444, 2025 Version 2. The final assessment was June 2025; the syllabus was available only to schools in Arizona, USA. This is not a current exam-entry option. Original self-check teaching and explained practice are not Cambridge papers, official marks or predicted grades. Linked teaching foundations do not establish full syllabus mastery. Calibrated cumulative assessment, reliable drawn-work review and qualified mathematics-teacher review remain unfinished.
Before you begin
- Sine values and degree mode
- Triangle angle sum
Work on paper. Try each question before revealing help, and explain why your method works.
Quick readiness check
What should you establish before applying the method in this lesson?
What you will learn
- Match each side with its opposite angle
- Complete a triangle from two angles and one side
- Reject angle sums that cannot form a triangle
Keep the opposite pairs together
With a opposite A, b opposite B and c opposite C, a/sinA=b/sinB=c/sinC. The historical 2025 syllabus supplies the sine-rule formula; choosing the correct pairs still matters. With two angles, first obtain the third from 180°−A−B. Each angle must be positive.
For A=30°, B=45° and a=10, C=105°. Then b=10sin45°/sin30°=10√2 and c=10sin105°/sin30°. These are ASA/AAS data, giving one triangle when the angle conditions are valid.
Calculate with precision
Use degree mode for numerical ratios, and keep intermediate sine values unrounded. SSA data give a different problem: inverse sine can produce two candidate angles, so that case needs the checks in the following lessons. Do not treat a missing or impossible third angle as a calculator-rounding problem.
Sources for this lesson
Pause and explain
Is a opposite A or adjacent to A in the sine-rule notation?
Worked example
For A=30°, B=45° and a=10 cm, find C and b.
Show the worked solution
- C=180°−30°−45°=105°.
- b=10sin45°/sin30°=10√2 cm.
Answer C=105°; b=10√2 cm.
Try a topic test
Choose Easy, Normal or Difficult. Save your answers and open worked review after finishing.
Test this topic →From guided practice to a new situation
Write your answer and reasoning first. Use a hint only if you are stuck. The model solution is for self-checking; your written work is not automatically marked.
A=30°, B=60° and a=5 cm. Find C, b and c exactly.
Give me a hint
Find the third angle and use the known opposite pair.
Compare my reasoning
- C=90°; b=5sin60°/sin30°=5√3 cm.
- c=5sin90°/sin30°=10 cm.
C=90°; b=5√3 cm; c=10 cm.
Look for these in your work
- Match opposite pairs
- Use exact special-angle values
A=45°, C=75° and c=12 m. Find B, a and b, giving lengths to 3 significant figures.
Give me a hint
Use c/sinC as the common ratio.
Compare my reasoning
- B=60°; a=12sin45°/sin75°≈8.784610 m.
- b=12sin60°/sin75°≈10.758906 m.
B=60°; a=8.78 m; b=10.8 m.
Look for these in your work
- Complete the angle sum
- Round only final lengths
A proposed triangle has A=40°, B=150° and a=10 cm. Can the sine rule produce a valid triangle?
Give me a hint
Check the third angle first.
Compare my reasoning
- The given angles total 190°.
- The required third angle would be negative, so no non-degenerate triangle exists.
No; the angle sum is already greater than 180°.
Look for these in your work
- Check feasibility before calculating
- Reject impossible angle data
Common mistakes
- Match a side to its opposite angle.
- Check the angle sum before using a formula.
What can you explain now?
Is a opposite A or adjacent to A in the sine-rule notation?
Compare with the explanation
Opposite A.
The side-angle pair must match across the triangle, not at the same corner.
After trying it yourself, choose your next review. This is your self-assessment.
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Make it stick
Is a opposite A or adjacent to A in the sine-rule notation?
If you needed the solution, close it and solve the problem again from a blank page. A correct answer is stronger when you can explain the reason for each step.