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Cambridge IGCSE Maths (0580) · Topic 6: Trigonometry · 6.5

Cambridge IGCSE Maths 0580 Sine and Cosine Rules Questions

Sine and Cosine Rules questions with worked answers for Cambridge IGCSE Mathematics (0580). Each question says whether it is Core or Extended and whether it is for a calculator or non-calculator paper. Try each one before you open the answer.

Practise sine and cosine rules →Sine and Cosine Rules notes

What the syllabus asks

Summarised in our own words: check the exact wording in the official 0580 syllabus.

Sine and Cosine Rules questions: Extended

Extended content, for Papers 2 and 4.

Question 1 Extended calculator · easy

In triangle \(ABC\), \(a = 7\) cm, \(b = 9\) cm and angle \(C = 40°\).
Work out the area of the triangle, correct to 3 significant figures.

Show the worked answer
  1. Area \(= \frac{1}{2}ab\sin C = \frac{1}{2} \times 7 \times 9 \times \sin 40° = 20.24\ldots\) cm²

Answer: \(20.2\) cm²

Question 2 Extended calculator · medium

In triangle \(ABC\), angle \(A = 42°\), angle \(B = 71°\) and \(a = 8\) cm.
Find \(b\), correct to 3 significant figures.

Show the worked answer
  1. Sine rule: \(\frac{b}{\sin 71°} = \frac{8}{\sin 42°}\)
  2. \(b = \frac{8 \sin 71°}{\sin 42°} = 11.30\ldots\) cm

Answer: \(11.3\) cm

Question 3 Extended calculator · medium

In triangle \(ABC\), angle \(A = 50°\), angle \(B = 60°\) and \(c = 10\) cm.
Find \(a\), correct to 3 significant figures.

Show the worked answer
  1. Angle \(C = 180° - 50° - 60° = 70°\).
  2. \(a = \frac{10 \sin 50°}{\sin 70°} = 8.152\ldots\) cm

Answer: \(8.15\) cm

Question 4 Extended calculator · medium

In triangle \(ABC\), \(b = 6\) cm, \(c = 9\) cm and angle \(A = 52°\).
Find \(a\), correct to 3 significant figures.

Show the worked answer
  1. Cosine rule: \(a^2 = 6^2 + 9^2 - 2 \times 6 \times 9 \cos 52° = 117 - 66.49\ldots = 50.50\ldots\)
  2. \(a = 7.106\ldots\) cm

Answer: \(7.11\) cm

Question 5 Extended calculator · medium

A triangle has sides \(5\) cm, \(7\) cm and \(9\) cm.
Find its largest angle, correct to 1 decimal place.

Show the worked answer
  1. The largest angle is opposite the \(9\) cm side.
  2. \(\cos\theta = \frac{5^2 + 7^2 - 9^2}{2 \times 5 \times 7} = \frac{-7}{70} = -0.1\)
  3. \(\theta = 95.73\ldots°\)

Answer: \(95.7°\)

Question 6 Extended calculator · hard

In triangle \(ABC\), \(a = 10\) cm, \(b = 14\) cm and angle \(A = 35°\).
Find the two possible values of angle \(B\), correct to 1 decimal place.

Show the worked answer
  1. \(\sin B = \frac{14 \sin 35°}{10} = 0.8030\ldots\)
  2. \(B = 53.42\ldots°\) or \(B = 180° - 53.42\ldots° = 126.57\ldots°\).
  3. Both are possible, because \(126.57\ldots° + 35° \lt 180°\) (the ambiguous case).

Answer: \(B = 53.4°\) or \(B = 126.6°\)

Question 7 Extended calculator · hard

A triangle has sides of \(8\) cm and \(10\) cm, and an area of \(30\) cm². The angle between these sides is acute.
Find this angle, correct to 1 decimal place.

Show the worked answer
  1. \(\frac{1}{2} \times 8 \times 10 \times \sin C = 30\), so \(\sin C = 0.75\).
  2. \(C = \sin^{-1} 0.75 = 48.59\ldots°\)

Answer: \(48.6°\)

Question 8 Extended calculator · hard

A boat sails \(12\) km on a bearing of \(050°\), then \(9\) km on a bearing of \(170°\).
How far is it from its starting point? Give your answer correct to 3 significant figures.

Show the worked answer
  1. At the turning point, the direction back to the start is \(230°\) and the new direction is \(170°\), so the angle between the two legs is \(60°\).
  2. \(d^2 = 12^2 + 9^2 - 2 \times 12 \times 9 \cos 60° = 225 - 108 = 117\)
  3. \(d = 10.81\ldots\) km

Answer: \(10.8\) km

Examiner Insights: common mistakes in IGCSE Maths 0580, with fixes (our analysis of public sources) →

Every question on this page is original, written for IGCSE Math Revision rather than taken from Cambridge papers, and each answer was re-solved independently before publishing. Questions with a mark scheme come from our 0580 practice bank (M method, A accuracy, B independent marks).

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Questions students ask

Which Cambridge 0580 papers can test sine and cosine rules?

Sine and Cosine Rules is Extended content (syllabus 6.5), so it is tested only on the Extended papers, Paper 2 (non-calculator) and Paper 4 (calculator). Core candidates do not need it.

Are these Cambridge past-paper questions?

No. Every question here was written for IGCSE Math Revision in the style of the 0580 papers, and each answer was checked independently before publishing. For real exam practice, use the Cambridge past papers linked from our 0580 past papers page.

Where can I practise more sine and cosine rules questions?

In the 0580 practice for topic 6 (Trigonometry), where every question is marked and tagged Core or Extended. Practice is part of the IGCSE plan, with a free preview; the questions on this page are free.

More trigonometry questions