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Geometry and measuresHigher

Sine and cosine rules GCSE questions with answers

8 sine and cosine rules questions to try, each with the answer and the full working one tap away. Every question here was made and worked out by the Show Working maths engine, not typed in by hand.

On Higher, the sine rule and the cosine rule find missing sides and angles in triangles without a right angle, and ½ab sin C finds the area. Which rule you use depends on which sides and angles you know.

Quiz yourself: 8 questions

Worked example 1: area of a triangle with ½ab sin c

Work out the area of the triangle. Give your answer to 1 decimal place.

11 cmC94°A9 cmB
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  1. You know two sides and the angle between them, so area = ½ × a × b × sin C.
  2. Area = ½ × 9 × 11 × sin 94° = 49.3794…
  3. To 1 decimal place: 49.4 cm².

Answer 49.4 cm²

01 / Quiz yourselfTry it, then tap to check

Sine and cosine rules questions to try

Work each one out on paper first. Then open the answer and compare your working with ours, step by step.

01 / Area of a triangle with ½ab sin CHigher only

The area of the triangle is 63 cm². Work out the length of the side marked x. Give your answer to 1 decimal place.

xD15 cmF70°E
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Show the answer and working
  1. The angle you know is between the side you know and x, so use area = ½ × a × b × sin C.
  2. ½ × 15 × x × sin 70° = 63
  3. x = 2 × 63 ÷ (15 × sin 70°) = 8.9391…
  4. To 1 decimal place: 8.9 cm.

Answer 8.9 cm

02 / Area of a triangle with ½ab sin CHigher only

Work out the area of the triangle. Give your answer to 1 decimal place.

10 cmK55°J6 cmL
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Show the answer and working
  1. You know two sides and the angle between them, so area = ½ × a × b × sin C.
  2. Area = ½ × 10 × 6 × sin 55° = 24.5746…
  3. To 1 decimal place: 24.6 cm².

Answer 24.6 cm²

03 / Area of a triangle with ½ab sin CHigher only

The area of the triangle is 37 cm². Work out the length of the side marked x. Give your answer to 1 decimal place.

105°K11 cmJxL
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  1. The angle you know is between the side you know and x, so use area = ½ × a × b × sin C.
  2. ½ × 11 × x × sin 105° = 37
  3. x = 2 × 37 ÷ (11 × sin 105°) = 6.9646…
  4. To 1 decimal place: 7.0 cm.

Answer 7.0 cm

04 / The sine ruleHigher only

Work out the length of the side marked x. Give your answer to 1 decimal place.

10 m26°BAx116°C
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  1. x is opposite the 116° angle and the 10 m side is opposite the 26° angle. Use the sine rule: x ÷ sin 116° = 10 ÷ sin 26°.
  2. x = 10 × sin 116° ÷ sin 26° = 20.5030…
  3. To 1 decimal place: 20.5 m.

Answer 20.5 m

05 / The sine ruleHigher only

Work out the length of the side marked x. Give your answer to 1 decimal place.

x74°AB8 cm28°C
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  1. x is opposite the 74° angle and the 8 cm side is opposite the 28° angle. Use the sine rule: x ÷ sin 74° = 8 ÷ sin 28°.
  2. x = 8 × sin 74° ÷ sin 28° = 16.3803…
  3. To 1 decimal place: 16.4 cm.

Answer 16.4 cm

06 / The sine ruleHigher only

Work out the length of the side marked x. Give your answer to 1 decimal place.

F10 cm27°Ex30°D
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  1. x is opposite the 30° angle and the 10 cm side is opposite the 27° angle. Use the sine rule: x ÷ sin 30° = 10 ÷ sin 27°.
  2. x = 10 × sin 30° ÷ sin 27° = 11.0134…
  3. To 1 decimal place: 11.0 cm.

Answer 11.0 cm

07 / The cosine ruleHigher only

Work out the size of the angle marked x. Give your answer to 1 decimal place.

12 mF10 mxD18 mE
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  1. You know all three sides, so use the cosine rule for an angle: cos A = (b² + c² − a²) ÷ 2bc, where a is the side opposite the angle.
  2. The side opposite x is 10 m. cos x = (12² + 18² − 10²) ÷ (2 × 12 × 18) = 368 ÷ 432 = 0.8519…
  3. x = cos⁻¹(0.8519…) = 31.5863…°
  4. To 1 decimal place: 31.6°.

Answer 31.6°

08 / The cosine ruleHigher only

Work out the length of the side marked x. Give your answer to 1 decimal place.

17 cmP14 cmRx45°Q
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  1. x is opposite the 45° angle, and you know the two sides either side of that angle. Use the cosine rule: a² = b² + c² − 2bc cos A.
  2. x² = 14² + 17² − 2 × 14 × 17 × cos 45°
  3. x² = 196 + 289 − 476 × cos 45° = 148.4172…
  4. x = √148.4172… = 12.1827…
  5. To 1 decimal place: 12.2 cm.

Answer 12.2 cm

02 / What's covered3 kinds of question

Sine and cosine rules at GCSE and where marks go

  • Area of a triangle with ½ab sin CHigher only
  • The sine ruleHigher only
  • The cosine ruleHigher only

Worked example 2: the cosine rule

Work out the length of the side marked x. Give your answer to 1 decimal place.

13 cmLx111°K20 cmJ
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  1. x is opposite the 111° angle, and you know the two sides either side of that angle. Use the cosine rule: a² = b² + c² − 2bc cos A.
  2. x² = 20² + 13² − 2 × 20 × 13 × cos 111°
  3. x² = 400 + 169 − 520 × cos 111° = 755.3513…
  4. x = √755.3513… = 27.4837…
  5. To 1 decimal place: 27.5 cm.

Answer 27.5 cm

A common mistake

Subtracting before multiplying in the cosine rule. In a² = b² + c² − 2bc cos A, work out 2bc cos A first. With b = 5, c = 8 and A = 60°, a² = 25 + 64 − 40 = 49, so a = 7.

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