Trigonometry — Free IB MYP5 Mathematics (Standard) Questions

1QuestionWorking with Two Sides and Included Angle SASConcept Practice

2 marks~3 minCriterion A

A surveyor measures two boundary lines from a fixed corner point

A

on a triangular plot of land:

AB = 8

cm and

AC = 5

cm on a scale drawing, with

\angle BAC = 60°

.

a

State the included angle between sides

AB

and

AC

. [1]

b

The surveyor needs to confirm whether boundary

BC

is shorter than 8 cm before placing a fencing order. Using the cosine rule with

\cos 60° = 0.5

, calculate

BC

, then advise the surveyor whether the fencing order should proceed. [1]

Solutions

2QuestionUsing Scale and Bearings TogetherConcept Practice

2 marks~3 minCriterion B

A coastguard station records four rescue vessels departing on different bearings.

Bearing	$030°$	$060°$	$120°$	$150°$
Clockwise angle from north	$30°$	$60°$	$120°$	$150°$

a

Describe the relationship between a bearing and its clockwise angle from north. [1]

b

A fifth vessel departs on a clockwise angle of

80°

from north. A safe shipping lane begins at a bearing of

080°

. Justify whether this vessel is travelling along the safe shipping lane, using bearing notation in your answer. [1]

Solutions

3QuestionUsing Scale and Bearings TogetherConcept Practice

4 marks~6 minCriterion A

A search-and-rescue team uses a map with a scale of

1:50\,000

. On the map, the straight-line distance between base camp

A

and a reported incident site

B

measures

8.4

cm. The bearing from

A

to

B

is

065°

.

a

Explain how the scale

1:50\,000

is used to convert the map distance to an actual ground distance. [1]

b

Determine the actual ground distance between

A

and

B

in kilometres. [2]

c

The rescue team can travel at most

5

km before requiring a resupply. Advise the team leader whether the team can depart from base camp

A

and reach site

B

without a resupply, justifying your advice with a numerical comparison. [1]

Solutions

4QuestionSketching Graphs of y = sin x cos x tan xConcept Practice

4 marks~6 minCriterion B

The graph of

y = \sin x

is shown. A second graph is drawn by adding 1 to each

y

-coordinate of

y = \sin x

, resulting in

y = \sin x + 1

.

Coordinates of key points on

y = \sin x

:
- Maximum:

(\pi/2, 1)

- Minimum:

(3\pi/2, -1)

- Zero at origin:

(0, 0)

a

Investigate how the

y

-coordinates change from

y = \sin x

to

y = \sin x + 1

for these points. Describe the pattern.

b

Generalize this pattern into a rule: what effect does adding a constant

c

to

\sin x

have on its graph?

c

Use your rule to predict the coordinates of the maximum and minimum points on

y = \sin x + 1

. Then verify by substituting

x = \pi/2

and

x = 3\pi/2

into

y = \sin x + 1

.

Solutions

5QuestionTransforming Trig Graphs AlgebraicallyConcept Practice

2 marks~3 minCriterion A

A buoy bobs in the ocean. Its vertical displacement from the water surface is modelled by

y = a \sin x

, where

x

represents time in seconds and

y

represents displacement in metres. The buoy reaches a maximum displacement of

3

m at

x = \dfrac{\pi}{2}

s and a minimum displacement of

-3

m at

x = \dfrac{3\pi}{2}

s.

a

Deduce the value of

a

. [1]

b

The buoy triggers a warning signal when its displacement exceeds

2.5

m. Justify whether the buoy activates the warning signal, using the model. [1]

Solutions

6QuestionUsing SOH-CAH-TOA to Find Missing SidesConcept Practice

4 marks~6 minCriterion A

A ladder leans against a wall. The diagram shows a right triangle where the ladder is 8 m long and makes a 35° angle with the ground. Study the diagram.

a

State which trigonometric ratio (SOH-CAH-TOA) links the angle, the hypotenuse, and the opposite side. [1 mark]

b

Calculate the height the ladder reaches up the wall. [1 mark]

c

Evaluate whether a ladder of the same length at 50° would reach more than 1 m higher, justifying your answer with calculations. [2 marks]

Solutions

7QuestionProblem Solving with Non-Right TrianglesConcept Practice

2 marks~3 minCriterion A

A landscape architect is designing a triangular garden plot. The plot is modelled as triangle

ABC

, where

AB = 8

m,

AC = 5

m, and the included angle at

A

is

30°

.

A local regulation states that any garden plot smaller than

12 \text{ m}^2

requires an additional permit.

a

Calculate the area of triangle

ABC

. [1]

b

Justify whether the architect must apply for an additional permit. [1]

Solutions

8QuestionSin 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

9QuestionSin and Cos rules based problemsConcept Practice

4 marks~6 minCriterion A

Triangle ABC has sides a = 8 cm, b = 11 cm, and angle A = 35°. Study the geometric figure.

a

State the formula for the Sine Rule as applied to triangle ABC. [1 mark]

b

Calculate angle B using the Sine Rule. [1 mark]

c

Evaluate whether a second valid solution for angle B exists, justifying your answer using properties of the sine function. [2 marks]

Solutions

10QuestionUsing Given Side and Opposite Angle SSAAssessment Practice

2 marks~3 minCriterion B

A lifeguard sits atop a chair with a seat height of 6 m above the water. The lifeguard observes a swimmer in distress at an angle of depression of 30° from horizontal. A pier extends 8 m horizontally from the base of the chair.

Using the sine rule for the triangle formed by the chair height, the pier, and the line of sight, determine whether one or two valid triangles exist for this SSA configuration. [1]

Advise the lifeguard whether the angle of depression alone is sufficient to locate the swimmer's position along the pier, justifying your answer with reference to your result. [1]

Solutions

11QuestionReal-Life Applications in Engineering and DesignAssessment Practice

4 marks~6 minCriterion D

A cable-stayed bridge has a vertical tower. A support cable runs from the top of the tower to a point on the deck. The cable makes an angle of elevation of

35°

with the horizontal. The horizontal distance from the base of the tower to the point where the cable meets the deck is

42

m.

a

Show that the height of the tower is given by

h = 42\tan(35°)

. [1]

b

Calculate the height of the tower, giving your answer to the nearest metre. [1]

c

A safety regulation states that the cable length must not exceed

1.5

times the tower height. Advise the bridge engineer whether this bridge satisfies the regulation, justifying your answer with a numerical comparison. [2]

Solutions

12QuestionUsing Given Side and Opposite Angle SSAAssessment Practice

3 marks~5 minCriterion C

A surveyor models a triangular plot of land as triangle

ABC

, where

a = 7

cm,

c = 5

cm, and

A = 70°

.

The graph below shows

y = \sin(x)

for

0° \leq x \leq 180°

, with a horizontal line drawn at

y = \dfrac{5\sin(70°)}{7} \approx 0.671

.

a

Identify the two possible values of angle

C

, rounding each to the nearest degree. [1]

b

Explain how the graph demonstrates that two distinct triangles are possible with the given information. [1]

c

The surveyor must choose the triangle in which angle

B

is obtuse. Justify whether this condition can be satisfied with the given measurements. [1]

Solutions

13QuestionDrawing Diagrams from Bearings DescriptionsAssessment Practice

6 marks~9 minCriterion D

A ship departs from port A and sails on a bearing of

060°

for 12 km to reach point B. It then changes course to a bearing of

150°

and travels 8 km to arrive at point C. A coastguard vessel at port A can only respond to emergencies within a 7 km radius.

a

Construct a diagram of the ship's journey. Include north lines at A, B, and C, and label all points and distances. [2]

b

Deduce the angle

\angle ABC

. [2]

c

The ship reports an emergency at point C. Advise the coastguard commander at A whether the vessel should be dispatched, justifying your answer with a calculated distance. [2]

Solutions

14QuestionDrawing Diagrams from Bearings DescriptionsAssessment Practice

8 marks~12 minCriterion C

A surveyor maps a closed path using four legs. The first three legs are:

Leg 1	bearing 030°	distance 5 km
Leg 2	bearing 150°	distance 5 km
Leg 3	bearing 270°	distance 5 km

a

Analyse the pattern in the bearings of Legs 1, 2, and 3, and deduce the bearing of Leg 4 that would continue this pattern to close the path. [2]

b

Calculate the resultant displacement after completing Legs 1, 2, and 3 by resolving each leg into its north and east components. [3]

c

Advise the surveyor whether Leg 4 should be included in the route plan, justifying your recommendation using your result from part (b) and the geometry of the path. [3]

Solutions

15QuestionTransforming Trig Graphs AlgebraicallyAssessment Practice

4 marks~6 minCriterion D

A coastal monitoring buoy records water depth. The depth, in metres, is modelled by

y = 3\sin(x) + 2

, where

x

is time in hours.

a

State the amplitude of

y = 3\sin(x) + 2

and explain what it represents in this context. [1]

b

Deduce the maximum and minimum water depths predicted by this model. [1]

c

Construct a clearly labelled sketch of

y = 3\sin(x) + 2

over one full period, then advise the harbour authority whether the harbour remains operational, given that a minimum water depth of 0 m is required at all times. [2]

Solutions

16QuestionSketching Graphs of y = sin x cos x tan xAssessment Practice

4 marks~6 minCriterion C

Given the trigonometric function

y = 3 \sin(2x)

, explain clearly, step-by-step, how to determine its amplitude, period, and principal axis. Then describe how to use these features to sketch one complete cycle of the graph, ensuring you use appropriate mathematical terminology and notation in your explanation.

Solutions

17QuestionUsing Trig Ratios to Find Missing AnglesAssessment Practice

12 marks~18 minCriterion B

A surveyor uses a drone to photograph a hillside. The drone's camera records the horizontal distance (adjacent side) and vertical rise (opposite side) for three sight lines from a fixed base point.

Sight line 1	opposite = 1 m	adjacent = 2 m
Sight line 2	opposite = 2 m	adjacent = 3 m
Sight line 3	opposite = 3 m	adjacent = 4 m

The angle of elevation

\theta

is measured from the horizontal at the base point.

a

Calculate

\theta_1

,

\theta_2

, and

\theta_3

, the angles of elevation for sight lines 1, 2, and 3. [3]

b

Deduce the angle of elevation

\theta_4

for a fourth sight line where opposite = 4 m and adjacent = 5 m, and justify whether the angles are increasing at a constant rate. [4]

c

The surveyor needs an angle of elevation of exactly 45° to calibrate the camera correctly. Advise the surveyor whether this calibration angle can ever be achieved using the pattern in this sequence, and recommend what alternative measurement setup would be required. [5]

Solutions

18QuestionDefining Sine Cosine and TangentAssessment Practice

2 marks~3 minCriterion D

A surveyor stands 40 m from the base of a telecommunications tower on level ground. Using a clinometer, she measures the angle of elevation to the top of the tower as 62°.

a

Explain how the tangent ratio is used to calculate the height of the tower, and state the height correct to one decimal place. [1]

b

Discuss one limitation of this method that affects the accuracy of the calculated height, and advise the surveyor whether she can confidently report that the tower meets a minimum height requirement of 75 m. [1]

Solutions

19QuestionUsing Trig Ratios to Find Missing AnglesAssessment Practice

3 marks~5 minCriterion C

A ladder leans against a vertical wall. The base of the ladder is 1.8 m from the wall, and the ladder reaches 3.0 m up the wall.

The graph below shows

y = \sin\theta

for

0° \leq \theta \leq 90°

.

a

Explain how to use the graph to find the angle

\theta

that the ladder makes with the ground. [1]

b

Determine the value of

\theta

. [1]

c

A safety guideline states that a ladder is stable when the angle with the ground is between 70° and 80°. Advise whether the ladder is positioned safely, justifying your answer with reference to

\theta

. [1]

Solutions

20QuestionFinding the Area of Any Triangle Using TrigonometryAssessment Practice

4 marks~6 minCriterion C

A landscape architect is designing a triangular garden bed with two fixed side lengths

a = 9.2

cm and

b = 12.5

cm (on a scale drawing). The graph below shows how the area of the triangle varies with the included angle

C

, using

\text{Area} = \frac{1}{2}ab\sin C

.

a

Show that the area formula simplifies to

\text{Area} = 57.5\sin C

. [1]

b

Explain how the shape of the graph shows that the maximum area occurs at

C = 90°

. [2]

c

The architect claims a right-angled triangular bed gives the most efficient use of the fixed side lengths. Calculate the maximum area correct to 3 significant figures, and advise the architect whether this right-angle design should be adopted to maximise the garden bed area. [1]

Solutions

21QuestionFinding the Area of Any Triangle Using TrigonometryAssessment Practice

8 marks~12 minCriterion D

A homeowner measures two sides of a triangular roof section for solar panel installation:

a = 6.8 \text{ m}

and

b = 5.2 \text{ m}

, with included angle

C = 72°

. The tape measure has accuracy

\pm 0.05 \text{ m}

and the protractor has accuracy

\pm 1°

.

\text{Area} = \tfrac{1}{2}ab\sin C

a

Calculate the area of the triangular roof section. [2]

b

Analyse how the measurement uncertainties affect the estimated area by calculating the maximum and minimum possible areas. [3]

c

Advise the homeowner whether the calculated area alone is sufficient to determine how many solar panels to order, identifying at least two limitations of the triangular model. [3]

Solutions

22QuestionFinding the Area of Any Triangle Using TrigonometryAssessment Practice

5 marks~8 minCriterion B

A city surveyor maps a triangular park with vertices at

A(0,0)

,

B(7,0)

, and

C(3,5)

, where units are in metres.

a

Show that

AB = 7

m,

AC = \sqrt{34}

m, and deduce that angle

A \approx 59.03°

. [2]

b

A fence post is to be placed at the point on

AB

directly below

C

. Calculate the area of the park using

\text{Area} = \frac{1}{2} \times \text{base} \times \text{height}

, identifying the perpendicular height clearly. [2]

c

The surveyor claims the trigonometric formula

\text{Area} = \frac{1}{2}\,ab\sin A

always gives the same area as the base–height method. Justify this claim algebraically, and explain what this means for calculating land areas when a perpendicular height cannot be measured directly. [1]

Solutions

23QuestionSin and Cos rules based problemsAssessment Practice

5 marks~8 minCriterion B

A surveying team is mapping a triangular plot of land,

ABC

. Their instruments record side

a = 8

cm, side

b = 10

cm, and angle

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

B

and side

c

. [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

24QuestionSin 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

Trigonometry — Free MYP5 Mathematics (Standard) Practice Questions