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23 Extend and Succeed Brain Growth â 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.
4â8 cm
48°
6â3 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.
15â3 cm
116°
10â7 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.
24â4 m 132°
18â4 m
88°
34â4 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 đĽ.
13. Triangle PQR is shown below.
If sin đ =1
4, calculate the area of the triangle.
Page | 9
14. Triangle ABC is shown below.
Given that sin đľ = â3
2,
show that the area of the triangle is â30 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.
â20 cm â8 cm
C
B
A
Page | 10
Practice Exam 1 Non-Calculator
1. Multiply out the brackets and collect like terms:
(2đĽ â 3)(đĽ2 â 2đĽ + 3) 3
2. Express đĽ2 + 6đĽ â 2 in the form (đĽ + đ)2 + đ. 2
3. Express â245 + â80 â â5 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
đŚ
đĽ
đŚ = 3đĽ â 2
2đĽ + đŚ = 8
Q
Page | 11
5. Expand and simplify đâ1
3 (đ1
3 + đ4
3). 2
6. Determine the nature of the roots of the equation
3đĽ2 â đĽ + 1 = 0. 3
7. Change the subject of the formula đ = âđđ
3 to đ. 2
8. Express 21
â7 with a rational denominator.
Give your answer in its simplest form. 2
9. Simplify đĽ2â9
đĽ2+8đĽ+15 . 3
10. The number of diagonals, đ, in a polygon with đ sides is given by the
formula đ =1
2đ(đ â 3).
(a) A polygon has 65 diagonals.
Show that for this polygon đ2 â 3đ â 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 â 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°
15â3 cm
116°
10â7 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 đ = 1
5 Calculate the length of side PR.
Leave your answer in the form âđ.
Page | 17
11. Triangle ABC is shown.
If cos đ´ = 0 â 5, show that
đĽ2 + 2đĽ â 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 đľ = 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 đ(đĽ) = 3đĽ + 2.
Given that đ(đ) = 23, calculate đ. 2
2. Express 5đĄ
đ á
đĄ
2đ ² in its simplest form. 3
3. There are 3 Ă 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 110Ë.
(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 đ such that the equation
đđĽÂ˛ â 2đĽ + 3 = 0, đ â 0, has no real roots. 4
7. Solve the equation 11 cos đĽË â 2 = 3, for 0 ⤠đĽ ⤠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°
3â8 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°
15â3 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 Taylorâs phone T, via a satellite
S, to Vickyâs phone V, forming a triangle STV as shown in the diagram.
(a) Calculate the distance from the satellite to Taylorâs 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.
5â1 m
11â4 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.
10â9 cm
7â6 cm
B
A
C 38°
Page | 29
Practice Exam 3 Non-Calculator
1. Multiply out the brackets and collect like terms
(đĽ â 4)(đĽ2 + đĽ â 2) 3
2. A straight line has equation 4đĽ + 3đŚ = 12.
(a) Find the gradient of this line. 2
(b) Find the coordinates of the point where this line crosses
the đĽ-axis. 2
3. Simplify
đĽÂ˛â4đĽ
đĽÂ˛+đĽâ20 3
4. Solve the equation
2đĽÂ˛ + 7đĽ â 15 = 0. 3
5. Change the subject of the formula đ = đđŁÂ˛
2 to đŁ. 3
6. A parabola has equation đŚ = đĽÂ˛ â 8đĽ + 19.
(a) Write the equation in the form đŚ = (đĽ â đ)2 + đ. 2
(b) Sketch the graph of đŚ = đĽÂ˛ â 8đĽ + 19, showing the
coordinates of the turning point and the point of
intersection with the đŚ-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 đŚ = đđ đđđđĽË is shown in the diagram.
State the values of đ and đ. 2
9. Solve algebraically the system of equations
3đĽ + 2đŚ = 17
2đĽ + 5đŚ = 4. 3
10. Express
4
đĽ+2 â
3
đĽâ4 đĽ â â2, đĽ â 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 4â6 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 13â5 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
4â8 metres and the
pupilâs eye level is
1â5 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, â, 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 65Ë, 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 ÂŁ98.
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 đŚ = đđĽÂ˛.
The point (2, -16) lies on the graph.
Determine the value of đ. 2
4. Solve the equation
2đĽÂ˛ + 3đĽ â 7 = 0.
Give your answers correct to 2 significant figures. 4
5. Prove that
sin2đ´
1âsin2đ´ = tan²đ´. 2
y
x
(2, -16)