0580/42 Feb/Mar 2023, Question 10: The Cosine Rule
Cambridge IGCSE Mathematics 0580 · Paper 42 (Extended) · Calculator · 15 marks
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The question, in short
Using quadrilateral ABCD with AC = 12.3 cm, AD = 16.5 cm, angle BAC = 31°, and right angles at B and C, work out lengths and angles involving AB, BD and the distance from C to BD.
Summarised in our own words. Use the official paper for the exact wording and the diagram.
Part (a) · 2 marks
Demonstrate that side AB equals 10.54 cm when rounded to two decimal places.
- 1cos 31° = AB/12.3
- 2AB = 12.3 × cos 31°
- 3AB = 10.543… = 10.54 cm (2 d.p.)
Answer10.543…
Part (b) · 2 marks
Demonstrate that angle DAC equals 41.80° when rounded to two decimal places.
- 1cos DAC = 12.3/16.5
- 2DAC = cos⁻¹(0.74545…)
- 3DAC = 41.801… = 41.80° (2 d.p.)
Answer41.801 to 41.802
Part (c) · 3 marks
Work out the length of diagonal BD.
- 1Angle BAD = 31 + 41.80 = 72.80°
- 2BD² = 10.54² + 16.5² − 2 × 10.54 × 16.5 × cos 72.80°
- 3BD² = 280.5…
- 4BD = 16.75 cm
Answer16.7 or 16.8 or 16.74 to 16.75…
Part (d) · 4 marks
Work out the size of angle CBD.
- 1BC = 12.3 sin 31° = 6.335, CD = 16.5 sin 41.80° = 10.998
- 2cos CBD = (16.75² + 6.335² − 10.998²)/(2 × 16.75 × 6.335)
- 3cos CBD = 0.9411…
- 4Angle CBD = 19.8°
Answer18.9 to 20.7…
Part (e) · 4 marks
Work out the least distance between point C and line BD.
- 1Shortest distance is the perpendicular from C to BD
- 2BC = 12.3 sin 31° = 6.335
- 3distance = 6.335 × sin 19.77°
- 4distance = 2.14 cm
Answer2.05 to 2.24…
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