Find an angle from three sides after checking triangle inequalities
Rearrange the cosine rule for an unknown angle · Use inverse cosine over 0° to 180° · Reject impossible or degenerate side data
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
- Cosine rule
- Strict triangle inequalities and inverse ratios
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
- Rearrange the cosine rule for an unknown angle
- Use inverse cosine over 0° to 180°
- Reject impossible or degenerate side data
Find the angle opposite the selected side
For angle C opposite c, cosC=(a²+b²−c²)/(2ab). In a valid triangle, inverse cosine selects the unique C between 0° and 180°. The largest side is opposite the largest angle. For a=5,b=7,c=8, the ratio is 1/7, giving C≈81.7868°.
Check before taking an inverse
Every side must be positive and each pair sum must exceed the remaining side. Equality gives a degenerate straight arrangement. Sides 2,3,5 fail the strict condition even though the formal cosine is −1; calling that a 180° triangle would hide the degeneracy. A computed ratio outside [−1,1] from exact data also signals that no real angle fits.
Sources for this lesson
Pause and explain
Do lengths 2,3,5 form a non-degenerate triangle?
Worked example
Find C for a=5 cm,b=7 cm,c=8 cm, to 1 decimal place.
Show the worked solution
- All strict triangle inequalities hold.
- cosC=(25+49−64)/70=1/7; use inverse cosine in degrees.
Answer 81.8°.
Try a topic test
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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.
Find C opposite side 5 in a triangle with a=3 cm,b=4 cm,c=5 cm.
Give me a hint
Rearrange the cosine rule.
Compare my reasoning
- cosC=(9+16−25)/24=0.
- C=90°.
90°.
Look for these in your work
- Match the opposite side
- Use the correct inverse
Find C opposite side 8 when the other sides are each 5 m, to 1 decimal place.
Give me a hint
Check feasibility, then calculate cosine.
Compare my reasoning
- 5+5>8 and the other inequalities hold.
- cosC=(25+25−64)/50=−0.28, so C≈106.2602°.
106.3°.
Look for these in your work
- Check the side conditions
- Recognise the obtuse result
Classify side sets (2,3,5) and (2,3,6) before attempting inverse cosine.
Give me a hint
Compare the two shorter sides with the longest.
Compare my reasoning
- For (2,3,5), equality makes the arrangement degenerate.
- For (2,3,6), the sum 5 is smaller than 6, so the data are impossible.
Neither is a non-degenerate triangle: the first is degenerate, the second impossible.
Look for these in your work
- Use strict inequalities
- Avoid forcing an inverse calculation
Common mistakes
- c must be the side opposite the angle being found.
- An equality case is not a non-degenerate triangle.
What can you explain now?
Do lengths 2,3,5 form a non-degenerate triangle?
Compare with the explanation
No; 2+3=5 is an equality case.
A triangle requires a strict greater-than inequality.
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Make it stick
Do lengths 2,3,5 form a non-degenerate triangle?
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.