Sine rule and opposite pairs
Match each angle to its opposite side and find a missing side in a non-right triangle.
Extended only. This is the separate 9–1 qualification 0980. Cambridge intends Core for grades 4–1 and Extended for grades 9–5; the eligible award bands are grades 5–1 for the Core papers and grades 9–3 for the Extended papers.
Before you begin
- Label triangle sides opposite capital-letter angles.
- Use sine in degree mode.
Work on paper. Try each question before revealing help, and explain why your method works.
Quick readiness check
Which angle is paired with side a?
What you will learn
- Choose the sine rule from an opposite side–angle pair.
- Calculate another side and check its size against the angles.
Keep opposite pairs together
In any triangle, a/sin A = b/sin B = c/sin C. Lower-case a lies opposite angle A, and similarly for b and c. This works for non-right triangles as well as right triangles. Draw the labels before substituting: adjacent sides do not make a sine-rule pair.
If a = 7 cm, A = 30° and B = 60°, then b = 7 sin 60°/sin 30° = 7√3 cm. The larger angle sixty has the longer opposite side. The third angle is 90°, from the triangle angle sum, but the sine-rule calculation did not require a right angle.
Decide what information is sufficient
A known opposite side–angle pair plus a second angle allows another side to be found. With two angles given, obtain the third from 180° if needed. Two sides and the included angle instead suggest the cosine rule because neither opposite pair is complete.
Write the equation with the unknown isolated, retain calculator precision and report a non-exact length to the requested accuracy. The longest side must lie opposite the largest angle. Finding an angle with inverse sine needs an extra check because sine can have an acute and an obtuse solution; that is developed in the ambiguous-case lesson.
Pause and explain
a = 7 cm, A = 30°, B = 60°. What is b?
Worked example
A triangle has a = 7 cm, A = 30° and B = 60°. Find b exactly.
Show the worked solution
- The known opposite pair is seven with thirty degrees.
- b = 7 sin 60°/sin 30° = 7(√3/2)/(1/2).
- Thus b = 7√3 cm, longer than a as B is larger than A.
Answer 7√3 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 = 5 cm, A = 30°, B = 90°. Find b.
Give me a hint
Use the known opposite pair five and thirty.
Compare my reasoning
- b/sin 90° = 5/sin 30°.
- b = 5 × 1/0.5 = 10 cm.
- The side opposite the largest angle is longest.
10 cm.
Look for these in your work
- Matched opposite pairs.
- Checked relative side sizes.
a = 10 cm, A = 35°, B = 42°. Find b to three significant figures.
Give me a hint
Isolate b before entering the degree-mode calculation.
Compare my reasoning
- b = 10 sin 42°/sin 35°.
- The unrounded result is about 11.666.
- Report 11.7 cm; forty-two is larger than thirty-five.
11.7 cm.
Look for these in your work
- Used degree mode.
- Rounded only the final result.
A triangle has A = 30°, B = 45° and a = 6 m. Find b exactly and the third angle.
Give me a hint
The angle sum and sine rule answer different parts.
Compare my reasoning
- C = 180 − 30 − 45 = 105°.
- b = 6 sin 45°/sin 30° = 6√2 m.
- Keep the surd exact; c would be longest because C is largest.
b = 6√2 m; C = 105°.
Look for these in your work
- Found the remaining angle.
- Retained the exact side value.
Common mistakes
- Pairing an angle with an adjacent side.
- Reversing the sine ratio.
- Assuming inverse sine always gives the only angle.
What can you explain now?
What information distinguishes a sine-rule pair?
Compare with the explanation
A side and its opposite angle.
Draw the side opposite the angle; an adjacent side does not belong to that pair.
After trying it yourself, choose your next review. This is your self-assessment.
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Make it stick
Tomorrow, label three opposite pairs and explain why the sine rule is appropriate before solving.
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.