Use the cosine rule with the included angle and correct sign
Identify the angle between the two known sides · Calculate the opposite third side · Handle negative cosine for an obtuse included angle
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
- Full-turn cosine values
- Squares and roots
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
- Identify the angle between the two known sides
- Calculate the opposite third side
- Handle negative cosine for an obtuse included angle
SAS determines one triangle
For sides a,b enclosing angle C, c²=a²+b²−2ab cosC. The historical 2025 syllabus gives the cosine-rule formula. Positive a,b and 0°<C<180° determine a non-degenerate triangle. The angle must be the one between those sides, not an arbitrary known angle elsewhere.
Subtract a negative correctly
For a=4, b=6, C=120°, cosine is −1/2, so c²=16+36−48(−1/2)=76. The subtraction adds 24, giving c=2√19. At C=90°, cosine is zero and the rule reduces to Pythagoras. At 0° or 180° the arrangement is degenerate rather than a valid triangle.
Sources for this lesson
Pause and explain
For obtuse C, does −2ab cosC add or subtract a positive amount?
Worked example
Find c when a=4 cm, b=6 cm and the included C=120°.
Show the worked solution
- c²=16+36−48(−1/2)=76.
- Take the positive root and keep the exact form.
Answer c=2√19 cm ≈ 8.72 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.
Find c for a=5 cm, b=8 cm and included C=60°.
Give me a hint
Use cos60°=1/2.
Compare my reasoning
- c²=25+64−80(1/2)=49.
- c=7 cm.
7 cm.
Look for these in your work
- Select the included angle
- Take a positive length
Find c for a=6 m, b=8 m and included C=120°, exactly and to 3 significant figures.
Give me a hint
Cosine is negative in quadrant two.
Compare my reasoning
- c²=36+64−96(−1/2)=148.
- c=2√37 m≈12.165525 m.
2√37 m ≈ 12.2 m.
Look for these in your work
- Keep the cosine sign
- Round only the final root
For a=3 cm and b=4 cm, compare C=90° with C=0°. Find the formal c values and decide which gives a non-degenerate triangle.
Give me a hint
Check the angle condition as well as the expression.
Compare my reasoning
- At 90°, c²=9+16=25 and c=5 cm.
- At 0°, c=1 cm, but 1+3=4 makes a straight, degenerate arrangement.
90° gives c=5 cm and a valid triangle; 0° gives c=1 cm but is degenerate.
Look for these in your work
- Check geometric feasibility
- Do not accept a formula result alone
Common mistakes
- Use the included angle.
- Subtracting a negative cosine contribution increases the side square.
What can you explain now?
For obtuse C, does −2ab cosC add or subtract a positive amount?
Compare with the explanation
It adds, because cosC is negative.
The two negative signs cancel in the cosine-rule term.
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Make it stick
For obtuse C, does −2ab cosC add or subtract a positive amount?
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.