Sin and Cos Rules
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Sin and Cos Rules — Free MYP4 Mathematics (Standard) Practice Questions

1QuestionSin and Cos rules based problemsConcept Practice
2 marks~3 minCriterion A
A surveyor measures a triangular plot of land, ABCABC. Angle A=35°A = 35°, angle B=72°B = 72°, and side a=8.2a = 8.2 cm (opposite angle AA).

Calculate the length of side bb, opposite angle BB, using the sine rule:

asin⁡A=bsin⁡B\frac{a}{\sin A} = \frac{b}{\sin B}

Give your answer in centimetres, correct to 3 significant figures. [2]
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Solutions

2QuestionSin and Cos rules based problemsAssessment Practice
2 marks~3 minCriterion D
A surveying team needs to find the straight-line distance across a river to a landmark CC on the opposite bank. Direct measurement is impossible.

The team stands at two points, AA and BB, on the same bank, measuring a baseline AB=84 mAB = 84\ \text{m}. They record the angle to the landmark from each point: ∠CAB=71°\angle CAB = 71° and ∠CBA=63°\angle CBA = 63°.

Apply the Sine Rule to find the distance ACAC, then advise the team on whether a rope of length 110 m110\ \text{m} is sufficient to reach the landmark from point AA. [2]
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Solutions

3QuestionSin and Cos rules based problemsAssessment Practice
8 marks~12 minCriterion D
Two search-and-rescue stations, A and B, are positioned 50 km apart along a straight coastline. Station A records the bearing of a distress signal as 30°30° from the coastline; station B records it as 50°50° from the coastline. Each bearing has an uncertainty of ±1°\pm 1° due to equipment limitations.
a
Show that the distance from station A to the signal is approximately 38.9 km, using the Law of Sines with exact bearings of 30°30° and 50°50°. [2]
b
Deduce the maximum possible range of distances from station A to the signal when the worst-case bearing errors of ±1°\pm 1° are applied simultaneously at both stations. [2]
c
A rescue coordinator must decide whether two-station triangulation is reliable enough to direct a rescue vessel in rough seas. Advise the coordinator whether this method should be used alone or combined with supplementary positioning, using your results from parts (a) and (b) and at least two real-world limitations of the model. [4]
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Solutions

4QuestionSin and Cos rules based problemsAssessment Practice
5 marks~8 minCriterion B
A surveying team is mapping a triangular plot of land, ABCABC. Their instruments record side a=8a = 8 cm, side b=10b = 10 cm, and angle A=35°A = 35°.
a
Deduce how many distinct triangles can be formed with these conditions, showing your use of the sine rule. [2]
b
For each possible triangle, calculate angle BB and side cc. [2]
c
The surveying team must fence only the plot with the larger perimeter. Advise the team which triangle to select, justifying your answer with a comparison of both perimeters. [1]

Solutions

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