Blue

23 Extend and Succeed Brain Growth – Senior Phase

Trigonometry

and bearings

Page | 2

Trig Rules

O1 I can use trigonometry to calculate area

1. ABC is a triangle as shown in the diagram with dimensions given.

Calculate the area of ABC to

the nearest square

centimetre.

2. Calculate the area of PQR, in square metres, to 2 significant figures.

4∙8 cm

48°

6∙3 cm

B

A

C

90 m

32°

120 m

R

P

Q

Page | 3

3. Chloe wishes to sow grass seed on a

triangular plot of ground.

The diagram gives the dimensions of

the plot.

Calculate the area of this plot to the

nearest square metre.

4. A farmer builds a sheep-pen using two lengths of fencing and a wall.

Calculate the area of the sheep-pen.

18 m

54°

32 m

Page | 4

5. A zoo plans to build an enclosure for some porcupines.

The regulations say that zoos should allow 20 square metres of

enclosure for each porcupine.

Will the enclosure shown be suitable for 4 porcupines?

Give a reason for your answer.

6. Calculate the area of PQR, in square metres, to 3 significant figures.

10 m 120°

12 m

63 m

58°

103 m

R

P

Q 38°

Page | 5

7. ABC is a triangle as shown in the diagram with dimensions given.

Calculate the area of ABC to

the nearest square

centimetre.

8. Paving stones are in the shape of a rhombus as shown.

Calculate the area of each paving stone.

15∙3 cm

116°

10∙7 cm

B

A

C 38°

Page | 6

9. A plot of ground, available for development, is in the shape of a

scalene quadrilateral.

The diagram below shows this plot of land with dimensions included.

The local authority building regulations state that the maximum

ground floor area of any new build has to be less than 80% of the

area of the plot.

Calculate the maximum ground floor area of any new build on this

plot to ensure it meets building regulations.

24∙4 m 132°

18∙4 m

88°

34∙4 m 20 m

Page | 7

10. PQR is the triangle shown below.

The area of the triangle PQR is 15 square centimetres.

Calculate the length of PQ.

11. ABC is the triangle shown below

The area of the triangle ABC is 12 square centimetres.

Angle BAC is obtuse.

Calculate the size of angle BAC.

6 cm

5 cm

C

A

B

Page | 8

12. ABC is an isosceles triangle aas shown.

The area of the triangle is 9 square centimetres.

Calculate the value of 𝑥.

13. Triangle PQR is shown below.

If sin 𝑃 =1

4, calculate the area of the triangle.

Page | 9

14. Triangle ABC is shown below.

Given that sin 𝐵 = √3

2,

show that the area of the triangle is √30 square centimetres.

15. A metal door-step is prism shaped, as shown.

The uniform cross-section is shown below.

Find the volume of metal required to make the door-step.

√20 cm √8 cm

C

B

A

Page | 10

Practice Exam 1 Non-Calculator

1. Multiply out the brackets and collect like terms:

(2𝑥 − 3)(𝑥2 − 2𝑥 + 3) 3

2. Express 𝑥2 + 6𝑥 − 2 in the form (𝑥 + 𝑝)2 + 𝑞. 2

3. Express √245 + √80 − √5 as a surd in its simplest form. 3

4. The graph below shows two straight lines.

The lines intersect at point Q.

Find algebraically, the coordinates of Q. 4

𝑦

𝑥

𝑦 = 3𝑥 − 2

2𝑥 + 𝑦 = 8

Q

Page | 11

5. Expand and simplify 𝑎−1

3 (𝑎1

3 + 𝑎4

3). 2

6. Determine the nature of the roots of the equation

3𝑥2 − 𝑥 + 1 = 0. 3

7. Change the subject of the formula 𝑅 = √𝑎𝑏

3 to 𝑎. 2

8. Express 21

√7 with a rational denominator.

Give your answer in its simplest form. 2

9. Simplify 𝑥2−9

𝑥2+8𝑥+15 . 3

10. The number of diagonals, 𝑑, in a polygon with 𝑛 sides is given by the

formula 𝑑 =1

2𝑛(𝑛 − 3).

(a) A polygon has 65 diagonals.

Show that for this polygon 𝑛2 − 3𝑛 − 130 = 0 2

(b) Hence find the number of sides in this polygon. 3

Page | 12

O2 The Cosine Rule

1. Triangle ABC is shown below.

Calculate the length of AB.

2. Triangle PQR is shown below.

Calculate the length of QR.

55 m

72°

120 m

R

P

Q

Page | 13

3. A wall is being built along one side of a triangular garden as shown.

Calculate the length of the wall.

4. A square trapdoor of side 80 centimetres is held open by a rod as

shown.

The rod is attached to a corner of the trapdoor and placed

40 centimetres along the edge of the opening.

The angle between the trapdoor and the opening is 76°.

Calculate the length of the rod to 2 significant figures

Page | 14

5. A telegraph pole, 6 ∙ 2 metres high, is blown over in the wind as

shown.

Calculate the length of AC.

6. The diagram shows a regular pentagon ABCDE.

EDF is a straight line.

(a) Write down the size of angle ABC.

(b) Calculate length of AC.

Page | 15

7. The diagram below shows triangle PQR.

Calculate the length of QR.

8. The diagram below shows triangle ABC.

Calculate the length of AB.

63 m

58°

103 m

R

P

Q 38°

15∙3 cm

116°

10∙7 cm

B

A

C 38°

Page | 16

9. As part of their training, netballers run around a triangular circuit

DEF shown below.

How many complete circuits must they run to cover at least

1000 metres.

10. Triangle PQR is shown below.

Given that cos 𝑄 = 1

5 Calculate the length of side PR.

Leave your answer in the form √𝑎.

Page | 17

11. Triangle ABC is shown.

If cos 𝐴 = 0 ∙ 5, show that

𝑥2 + 2𝑥 − 12 = 0

12. Triangle DEF is shown below.

Calculate the size of angle EDF.

Page | 18

13. The triangle below show the Bermuda Triangle an area in the

Atlantic.

Its vertices are at Bermuda (B), Miami (M) and Puerto Rico (P).

Calculate the size of angle BPM.

14. Triangle PQR is shown below.

Calculate the size of angle QPR.

Page | 19

15.

In triangle ABC, show that cos 𝐵 = 5

9

16. A table top is fixed to the legs of the table by a hinge.

The diagram below shows the table top, legs and hinge with

dimensions given.

The hinge is set to an obtuse angle.

Calculate the size of this angle.

Page | 20

17. Quadrilateral ABCD is shown below.

(a) Calculate the length of AC.

(b) Calculate the size of angle ADC.

Page | 21

Practice Exam 2 Calculator

1. A function is defined as 𝑓(𝑥) = 3𝑥 + 2.

Given that 𝑓(𝑎) = 23, calculate 𝑎. 2

2. Express 5𝑡

𝑠 ÷

𝑡

2𝑠² in its simplest form. 3

3. There are 3 × 105 platelets per millilitre of blood.

On average, a person has 5·5 litres of blood.

On average, how many platelets does a person have in

their blood?

Give your answer in scientific notation. 2

4. A child's toy is in the shape of a

hemisphere with a cone on top, as

shown in the diagram.

The toy is 12 centimetres wide

and 17 centimetres high.

Calculate the volume of the toy.

Give your answer correct to 2 significant figures. 5

12cm

17cm

Page | 22

5. A cone is formed from a paper circle with a sector removed as

shown.

The radius of the paper circle is 40 centimetres.

Angle AOB is 110˚.

(a) Calculate the area of the sector removed from the

circle. 3

(b) Calculate the circumference of the base of the

cone. 3

6. Find the range of values of 𝑝 such that the equation

𝑝𝑥² − 2𝑥 + 3 = 0, 𝑝 ≠ 0, has no real roots. 4

7. Solve the equation 11 cos 𝑥˚ − 2 = 3, for 0 ≤ 𝑥 ≤ 360. 3

110°

40cm

O

B

A

Page | 23

O3 The Sine Rule

1. ABC is a triangle as shown in the diagram with dimensions given.

Find the length of side CB.

2. Calculate the length of PR in centimetres, to 3 significant figures.

58°

3∙8 cm

B

A

C 53°

20 cm

25°

R

P

Q

79°

Page | 24

3. The diagram below shows triangle PQR.

Calculate the length of QR.

4. The diagram below shows triangle ABC.

Calculate the length of AB.

63 m

58°

R

P

Q 38°

15∙3 cm

116°

B

A

C 38°

Page | 25

5. A cable CB is connected to a vertical wall.

The angle between CB and the horizontal is 22°.

A second cable AB is also connected to the same wall and is 8 metres

long.

The angle between CB and the horizontal is 59°.

Calculate the length of cable CB.

6.

(a) Calculate the length of TG.

(b) Calculate the length of TB.

Page | 26

7. A Helicopter, at point H, hovers between two boats at points A and B

as shown in the diagram.

Calculate the distance from the helicopter to the nearer boat.

8. A mobile phone signal is sent from a Taylor’s phone T, via a satellite

S, to Vicky’s phone V, forming a triangle STV as shown in the diagram.

(a) Calculate the distance from the satellite to Taylor’s phone (ST).

(b) Assuming that the side of the triangle TV is horizontal, calculate

the height of the satellite above the ground.

Page | 27

9. The diagram below shows triangle PQR.

Calculate the size of acute angle QRP.

10. Triangle ABC is shown below

Find the size of angle BAC.

5∙1 m

11∙4 m

R

P

Q 23°

Page | 28

11. Triangle ABC is given below.

Find the size of angle ABC.

12. The diagram below shows triangle ABC.

Calculate the size of obtuse angle CAB.

10∙9 cm

7∙6 cm

B

A

C 38°

Page | 29

Practice Exam 3 Non-Calculator

1. Multiply out the brackets and collect like terms

(𝑥 − 4)(𝑥2 + 𝑥 − 2) 3

2. A straight line has equation 4𝑥 + 3𝑦 = 12.

(a) Find the gradient of this line. 2

(b) Find the coordinates of the point where this line crosses

the 𝑥-axis. 2

3. Simplify

𝑥²−4𝑥

𝑥²+𝑥−20 3

4. Solve the equation

2𝑥² + 7𝑥 − 15 = 0. 3

5. Change the subject of the formula 𝑝 = 𝑚𝑣²

2 to 𝑣. 3

6. A parabola has equation 𝑦 = 𝑥² − 8𝑥 + 19.

(a) Write the equation in the form 𝑦 = (𝑥 − 𝑝)2 + 𝑞. 2

(b) Sketch the graph of 𝑦 = 𝑥² − 8𝑥 + 19, showing the

coordinates of the turning point and the point of

intersection with the 𝑦-axis. 3

Page | 30

30 60 90 120 150 180 210 240 270 300 330 360

-4

4

x

y

7. Find the equation of the line joining the points (-2, 5)

and (3, 15).

Give the equation in its simplest form. 3

8. Part of the graph of 𝑦 = 𝑎𝑠𝑖𝑛𝑏𝑥˚ is shown in the diagram.

State the values of 𝑎 and 𝑏. 2

9. Solve algebraically the system of equations

3𝑥 + 2𝑦 = 17

2𝑥 + 5𝑦 = 4. 3

10. Express

4

𝑥+2 −

3

𝑥−4 𝑥 ≠ −2, 𝑥 ≠ 4

as a single fraction in its simplest form. 3

Page | 31

O4 Bearings and Trigonometry

1. (a) There are three mooring points A, B and C on Lake Kilbride.

From A, the bearing of B is 074°.

From C, the bearing of B is 310°.

Calculate the size of angle ABC.

(b) B is 230 metres from A and 110 metres from C.

Calculate the direct distance from A to C.

Give your answer to 3 significant figures.

Page | 32

2. Two yachts leave from harbour H as shown in the diagram.

Yacht A sails for 30km on a bearing of 072° and stops.

Yacht B sails for 50km on a bearing of 140° and stops.

How far apart are the two yachts when they stop.

3. David walks on a bearing of 050° from hostel A to a viewpoint V,

5 kilometres away.

Hostel B is due east of hostel A.

Susie walks on a bearing of 294° from hostel B to the same viewpoint.

Calculate the length of AB, the distance between the two hostels.

Page | 33

4. Brunton is 30 kilometres

due North of Appleton.

From Appleton, the bearing

of Carlton is 065°.

From Brunton, the bearing

of Carlton is 153°.

Calculate the distance between

Carlton and Brunton.

5. Jane is taking part in an orienteering competition.

She should have run 160 m from A to B on a bearing of 032°.

However, she actually ran 160 m from A to C on a bearing of 052°.

(a) Write down the size of angle BAC.

(b) Calculate the length of BC.

(c) What is the bearing from C to B.

Page | 34

6. Three radio masts, Kangaroo (K), Wallaby (W) and Possum (P) are

situated in the Australian outback as shown in the diagram.

Kangaroo is due south of Wallaby.

Possum is on a bearing of 130° from Kangaroo.

Calculate the bearing of Possum from Wallaby.

Page | 35

O5 Angles of Elevation or Depression

1. A statue stands in the corner of a square courtyard.

The statue is 4∙6 metres high.

The angle of elevation from the opposite corner to the top of the

statue is 8°.

Find the length of the diagonal of the courtyard.

2. A boat is 20 metres directly above sunken treasure.

A diver, on the surface of the water, has fixed a tight line to the

treasure at an angle of depression of 28° using some rope.

What is the length of this rope?

28°

20 metres

Page | 36

3. A lamp-post, CT, is supported by two wires AT and BT as shown in the

diagram. The lamp-post is 13∙5 metres high.

A, B and C all lie on the same horizontal piece of ground.

AT is set at an angle of elevation of 40°.

BT is set at an angle of elevation of 70°.

(a) Calculate the distance from B to C.

(b) Calculate the distance from A to B.

4. A balloon, position B, is attached

by wires, A and C, to the ground

as shown in the diagram.

From A, the angle of elevation to

B is 53°.

From C, the angle of elevation to

B is 68°.

Calculate the height of the balloon above the ground.

Page | 37

5. Two ships have located a wreck on the sea bed.

In the diagram below, the points P and Q represent the two ships and

the point R represents the wreck.

The angle of depression of R from P is 27°.

The angle of depression of R from Q is 42°.

Calculate QS, the distance ship Q must travel to be directly above

the wreck.

6. The diagram shows two positions of a pupil viewing the top of a tower

block.

From A, the angle of

elevation to T is 69°.

From B, the angle of

elevation to T is 64°.

The distance AB is

4∙8 metres and the

pupil’s eye level is

1∙5 metres.

Find the height of the tower block.

Page | 38

7. The diagram shows two blocks of flats of equal height.

A and B represent points on the top of the flats and C represents a

point on the ground between them.

The angle of depression from A to C is 38°.

The angle of depression from B to C is 46°.

Calculate the height of each block of flats, ℎ, in metres.

8. For safety reasons a building is supported by two struts represented

in the diagram below by DB and DC.

BD is set at an angle of elevation of 55°.

CD is set at an angle of elevation of 38°.

Calculate the height of the building represented by AD.

Page | 39

9. An aeroplane is flying so AB is parallel to the ground.

Lights A and B have been fitted so that they meet exactly on

the ground at C at a certain height.

The angle of depression of the beam of light from A to C at

the certain height is 50°.

The angle of depression of the beam of light from B to C at

the certain height is 70°.

AB is 20 metres in length.

Find the height of the aeroplane above C.

Page | 40

Practice Exam 4 Calculator

1. The pendulum of a antique clock swings along an arc of a circle,

centre O.

The pendulum swings through an angle of 65˚, travelling

from A to B.

The length of the arc AB is 32·6 centimetres.

Calculate the length of the pendulum. 4

2. Two groups of people go to a theatre.

Bill buys tickets for 5 adults and 3 children.

The total cost of his tickets is £158·25.

(a) Write down an equation to illustrate this information. 1

(b) Ben buys tickets for 3 adults and 2 children.

The total cost of his tickets is £98.

Write down an equation to illustrate this information. 1

(c) Calculate the cost of a ticket for an adult and the cost

of a ticket for a child. 4

65°

A B

O

Page | 41

3. The graph shown has an equation in the form 𝑦 = 𝑘𝑥².

The point (2, -16) lies on the graph.

Determine the value of 𝑘. 2

4. Solve the equation

2𝑥² + 3𝑥 − 7 = 0.

Give your answers correct to 2 significant figures. 4

5. Prove that

sin2𝐴

1−sin2𝐴 = tan²𝐴. 2

y

x

(2, -16)