Solve non-right-angled triangle problems
Use the sine rule, cosine rule and ½ab sin C to solve any triangle, including obtuse and ambiguous cases.
Section 7 · Trigonometry
Use this pathway for Cambridge IGCSE International Mathematics 0607, 2028–2030. Core and Extended objectives are labelled in each lesson; Extended adds requirements beyond Core. Paper 1 and Paper 2 are non-calculator. Papers 3–6 require a graphic display calculator (GDC). These original lessons use self-checked practice; they are not a Cambridge grade, full timed paper or teacher review.
Before you begin
- Check whether your school has entered you for Core or Extended. Core and Extended share most Number ideas, but some notation and methods are Extended only.
Work on paper. Try each question before revealing help, and explain why your method works.
Quick readiness check
In triangle notation, which angle is opposite side c?
What you will learn
- Choose the sine rule or cosine rule from known side-angle pairs
- Find triangle area from two sides and their included angle
- Recognise both valid solutions in the ambiguous sine-rule case
Match opposite pairs before using the sine rule
a/sin A = b/sin B = c/sin C. Each side letter is opposite the matching capital angle. Use the sine rule when you know an opposite side-angle pair and another side or angle.
For a side-side-angle (SSA) case, the inverse sine can produce θ or 180°−θ. Check whether both make the angle sum less than 180° and whether both fit the given data.
Use the cosine rule when two sides enclose an angle
c² = a² + b² − 2ab cos C, with C between sides a and b. For an obtuse C, cos C is negative, so the minus-negative term increases c².
Triangle area = ½ab sin C, where C is the included angle. These three formulas are supplied in the syllabus; show which pair or included angle you used.
Pause and explain
When is the cosine rule useful for finding a third side?
Worked example
Two sides of a triangle are 7 cm and 9 cm with included angle 60°. Find the third side and area exactly.
Show the worked solution
- Cosine rule: c² = 7² + 9² − 2(7)(9)cos 60° = 49 + 81 − 63 = 67.
- So c = √67 cm.
- Area = ½(7)(9)sin 60° = 31.5 × √3/2 = 63√3/4 cm².
Answer Third side √67 cm; area 63√3/4 cm².
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.
In triangle ABC, a = 8 cm, A = 30° and B = 45°. Find b exactly.
Give me a hint
Match each side to its opposite angle in the sine rule.
Compare my reasoning
- a/sin A = b/sin B, so b = 8 sin 45°/sin 30°.
- b = 8(√2/2)/(1/2).
- b = 8√2 cm.
8√2 cm.
Look for these in your work
- Opposite pairs matched
- Sine rule rearranged
- Exact value retained
Two sides of a triangle are 5 cm and 8 cm with included angle 120°. Find the third side to 3 significant figures.
Give me a hint
Use the cosine rule and remember cos 120° is negative.
Compare my reasoning
- c² = 5² + 8² − 2(5)(8)cos 120°.
- c² = 89 − 80(−1/2) = 129.
- c = √129 ≈ 11.4 cm to 3 s.f.
11.4 cm.
Look for these in your work
- Included angle identified
- Obtuse cosine sign handled
- Final length rounded to 3 s.f.
A triangle has sides 10 cm and 12 cm with included angle 30°. Find its exact area and explain which measurements are needed.
Give me a hint
Use half the product of the two sides and the sine of the angle between them.
Compare my reasoning
- The included angle is 30°, so area = ½(10)(12)sin 30°.
- sin 30° = 1/2.
- Area = 60 × 1/2 = 30 cm².
30 cm²; use two sides and their included angle.
Look for these in your work
- Correct angle selected
- Area formula applied
- Square units included
Common mistakes
- Pairing a side with an angle that is not opposite it
- Forgetting the included-angle condition for area
- Discarding a second valid SSA angle
- Using a rounded angle too early
What can you explain now?
Which formula uses two sides and their included angle to find area?
Compare with the explanation
Area = ½ab sin C
C is the angle between the two sides a and b.
After trying it yourself, choose your next review. This is your self-assessment.
Your review choice appears on Today. Sign in to sync it across devices.
Make it stick
How do the known side-angle pairs tell you whether the sine rule or cosine rule is the better choice?
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.