Sin and Cos Rules

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

1QuestionSin and Cos rules based problemsConcept Practice

2 marks~3 minCriterion A

A surveyor measures a triangular plot of land,

ABC

. Angle

A = 35°

, angle

B = 72°

, and side

a = 8.2

cm (opposite angle

A

).

Calculate the length of side

b

, opposite angle

B

, using the sine rule:

\frac{a}{\sin A} = \frac{b}{\sin B}

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

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

C

on the opposite bank. Direct measurement is impossible.

The team stands at two points,

A

and

B

, on the same bank, measuring a baseline

AB = 84\ \text{m}

. They record the angle to the landmark from each point:

\angle CAB = 71°

and

\angle CBA = 63°

.

Apply the Sine Rule to find the distance

AC

, then advise the team on whether a rope of length

110\ \text{m}

is sufficient to reach the landmark from point

A

. [2]

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°

from the coastline; station B records it as

50°

from the coastline. Each bearing has an uncertainty of

\pm 1°

due to equipment limitations.

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°

and

50°

. [2]

Deduce the maximum possible range of distances from station A to the signal when the worst-case bearing errors of

\pm 1°

are applied simultaneously at both stations. [2]

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]

Solutions

4QuestionSin and Cos rules based problemsAssessment Practice

8 marks~12 minCriterion B

A surveying team records two side lengths and a non-included angle for four separate plots of land. For each plot, the team must determine how many distinct triangular boundaries are geometrically possible before construction can begin.

Plot A	$a = 8$ cm	$b = 12$ cm	$A = 30°$
Plot B	$a = 10$ cm	$b = 8$ cm	$A = 40°$
Plot C	$a = 6$ cm	$b = 10$ cm	$A = 25°$
Plot D	$a = 7$ cm	$b = 7$ cm	$A = 50°$

Here

a

is the side opposite angle

A

, and

b

is the other given side.

Calculate all possible values of angle

B

for each plot using the sine rule. State the number of distinct triangles possible in each case. [4]

Deduce a general rule, in terms of

a

b

, and

A

, that predicts when exactly two distinct triangles are possible. [2]

A construction permit is issued only when a unique triangular boundary exists. Justify which plots receive a permit, using your rule from part (b) and the geometry of the sine ratio to explain why ambiguous cases must be rejected. [2]

Solutions

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