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AS Pure Mathematics · Edexcel 8MA0 · free A-level maths worksheet

Sine and cosine rules

Sine rule, cosine rule, area = half ab sin C, the ambiguous case, problems in context.

12 original questions, 53 marks. Print the worksheet, mark it with the answers sheet, and use the solutions to see every step written out by hand.

Worksheet PDFWorked solutions PDFAnswers PDF

The questions

1

The diagram shows triangle ABC, with AB = 7 cm, AC = 10 cm and angle BAC = 50°

Find the length of BC. Give your answer to 3 significant figures.

10 cm7 cm50°BCADiagram NOT accurately drawn
3 marks
Show answer
7.68 cm
2

In triangle ABC, AB = 12 cm, angle ABC = 47° and angle ACB = 58°

Find the length of AC. Give your answer to 3 significant figures.

12 cm47°58°BCADiagram NOT accurately drawn
3 marks
Show answer
10.3 cm
3

Triangle ABC has AB = 5 cm, AC = 8 cm and BC = 10 cm.

Find the size of the largest angle in the triangle. Give your answer in degrees to 1 decimal place.

10 cm8 cm5 cmBCADiagram NOT accurately drawn
3 marks
Show answer
97.9°
4

The diagram shows triangle ABC with AB = 6.4 cm, AC = 8.5 cm and angle BAC = 112°

Find the area of the triangle. Give your answer to 3 significant figures.

8.5 cm6.4 cm112°BCADiagram NOT accurately drawn
2 marks
Show answer
25.2 cm2
5

In triangle ABC, AB = 14 cm, BC = 9 cm and angle ABC = 63°

  1. Find the length of AC, giving your answer to 3 significant figures.
  2. Find the size of angle BAC, giving your answer to 1 decimal place.
9 cm14 cm63°BCADiagram NOT accurately drawn
5 marks
Show answer
a) 12.8 cm b) 39.0°
6

In triangle ABC, AB = 11 cm, BC = 8 cm and angle BAC = 35°. The diagram shows one possible shape for the triangle.

Find the two possible values of angle ACB, giving your answers to 1 decimal place.

11 cm8 cm35°ABCDiagram NOT accurately drawn
4 marks
Show answer
52.1° or 127.9°
7

Triangle ABC has area 30 cm2. AB = 8 cm and angle BAC = 40°

  1. Find the length of AC.
  2. Find the length of BC.

Give your answers to 3 significant figures.

8 cm40°BCADiagram NOT accurately drawn
5 marks
Show answer
a) 11.7 cm b) 7.56 cm
8

The diagram shows triangle ABC with AB = x cm, AC = (x + 2) cm, BC = 7 cm and angle BAC = 120°

  1. Show that x2 + 2x − 15 = 0
  2. Hence find the value of x.
  3. Find the exact area of the triangle.
7 cm(x + 2) cmx cm120°BCADiagram NOT accurately drawn
6 marks
Show answer
a) 72 = x2 + (x + 2)2 - 2x(x + 2)cos 120° leads to x2 + 2x - 15 = 0 b) x = 3 c) 15√3/4 cm2
9

A boat leaves harbour A and sails 12 km on a bearing of 050° to a buoy B. It then sails 8 km on a bearing of 150° to a lighthouse C.

  1. Show that angle ABC = 80°
  2. Find the distance AC, giving your answer to 3 significant figures.
  3. Find the bearing of C from A, giving your answer to the nearest degree.
12 km8 km050°150°ABCNNDiagram NOT accurately drawn
6 marks
Show answer
a) bearing of A from B is 230°, 230° - 150° = 80° b) 13.2 km c) 087°
10

A farmer's field is in the shape of triangle ABC, with AB = 120 m, BC = 95 m and AC = 150 m.

  1. Find the size of angle ABC, giving your answer to 1 decimal place.
  2. Find the area of the field in hectares, where 1 hectare = 10 000 m2. Give your answer to 2 significant figures.
95 m150 m120 mBCADiagram NOT accurately drawn
5 marks
Show answer
a) 87.7° b) 0.57 hectares
11

Triangle ABC has AB = 6 cm, BC = 4 cm and AC = 5 cm.

  1. Show that cos ACB = 18
  2. Hence find the exact area of triangle ABC, giving your answer in the form k√7, where k is a rational number.
4 cm5 cm6 cmBCADiagram NOT accurately drawn
5 marks
Show answer
a) cos C = (16 + 25 - 36)/40 = 5/40 = 1/8 b) (15/4)√7 cm2
12

In triangle ABC, AB = 10 cm, AC = 6 cm and angle ABC = 30°. The length of BC is a cm. The diagram shows one possible shape for the triangle.

  1. Show that a2 − 10√3a + 64 = 0
  2. Hence find the two possible exact lengths of BC.
  3. Explain why there are two possible triangles.
a cm6 cm10 cm30°BCADiagram NOT accurately drawn
6 marks
Show answer
a) 62 = 102 + a2 - 2 × 10 × a × cos 30° gives a2 - 10√3a + 64 = 0 b) 5√3 + √11 or 5√3 - √11 c) Both roots are positive, so both give a triangle: AC (6 cm) is longer than the distance from A to line BC (5 cm) but shorter than AB

Every question here is original and written by Robinson Tuition. It follows the Edexcel AS specification and suits AQA and OCR AS maths too; this is not an exam-board paper.

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