Mathematics: Analysis and Approaches

Exercise 1iG 4 Find two possible values for CA in

It might help you to draw a rough sketch triangle ABC when BA = 400, b = 24 and

to view these triangles. You know that this c = 30.

triangle might have an ambiguous case since S When the area of this triangle is 20 cm2,
you are given two sides and an angle that is
find the size of obtuse angle ABC.
not in between them. In exams, it is common
8 cm 10cm
to be asked for two possible values. c

1 Find two possible values for Ä in triangle
ABC when O = 640, a = 10 and c = 8.

2 Find two possible values for A in triangle
ABC when 200, b = 3 and a = 5.

3 Find two possible values for Bin triangle

ABC when 450, a = 8 and b = 10.

11.4 The cosine rule

The Pythagorean theorem and right-angle triangle relationships are the

foundations of trigonometry. In the previous section, you saw how the

sine rule could be derived using right-angle triangles. The sine rule can

be used when you have an angle, its opposite side, and one more piece
of information, but not when you have two sides and the included

angle. The Pythagorean theorem and right-angle triangle trigonometry
come to the rescue again, to find a suitable method for this situation.

Investigation 10

You will be using the triangle. You might want to work with a friend to compare results.
COnce tual How does the relationship between the sides and the angles of triangles lead
you to the cosine rule?

2 In triangle ADC, apply the Pythagorean theorem to c a

complete the equation b2 = b B
A3 Still using triangle ADC, complete cos = ... and then
D 489
rewrite this equation to make x the subject. x

4 Now use the Pythagorean theorem in triangle BCD to

complete the equationa2 =

S Expand the expression (c — x)2
You should now have a2 = c2 — 2cx+x2 + h2.

6 Substitute your answer from question 2.

7 Substitute your answer from question 3.

8 Simplify.

You now have the cosine rule.

11 RELATIONSHIPS IN SPACE: GEOMETRY AND TRIGONOMETRY IN 20 AND 3D

AABCThe cosine rule to find a side in is

a2 = b2 + ca — 2bc cosA

or b2 cosB or a2 +b2 — 2ab cosC

The cosine rule may also be used to find an angle when you are

given three sides.

The cosine rule to find an angle in LABC is

2

cosA =

2bc

or cosB = — 112

2ac

or cosC =

2ab

Example 11 b Find A.

a Find BC:

17.9 cm 620 27 cm

14.7 cm 18 cm

c 21 cm c

a a2 = + c2 — 2bc cosA 14.7, 17.9, 620
BC2 = 14.72 + 17.92 -
cos62 Substitute into the cosine rule.
BC = 17.Ocm
Evaluate.
b2+C —a2
b COSA = a = 18, b=21, 27

2bc Substitute into the cosine rule.
Evaluate.
212 +272-182
COSA =

2x21x27

846

A = = 41.80

1134

490

11.4

Exercise 1iH

1 Find the missing side lengths. 2 Find 0:
a
a
620 10.4 cm 18 cm
9 cm
12 cm

21.9 cm

c b

bA 8.6 3.1 cm

28 cm c 9.7 cm
65 cm
1120 118 cm
B 15 cm c d 9
c 3 cm 55 cm

5 cm

c

22 m

14 m

800 5 cm

d

20 cm 24 cm

10m

660

c 22 cm
e
3.8 cm 4 cm
25 cm

200

40 cm

c 3 The lengths of the three sides of a triangle
c
are 6cm, 9cm and 12 cm.
21m
a Find the measure of the smallest angle
1230 in the triangle.

30 m b Find the area of the triangle.

491

11 RELATIONSHIPS IN SPACE: GEOMETRY AND TRIGONOMETRY IN 20 AND 3D

4 A ship leaves port at 2pm traveling north 6 A lighthouse keeper spots a yacht 15 km

at a speed of 30 kmh-l. At 4pm, the ship away on bearing 0700. There is also

Howadjusts its course 200 to the east. far, a rock 9 km from the lighthouse on a
bearing of 2100. How far is the yacht from
to the nearest km, is the ship from the port
the rock?
at 5pm?
North
S Joon stands between two buildings of
Yacht
mdifferent heights. He is 46 away from
15m
the building on his left, and the angle of
9km
elevation to the top of the building is 330.

mThe building on his right is 28 away, with

an angle of elevation of 170 to the top of

the building.

a Find the heights of the two buildings.

b Find the distance from the top of one
building to the top of the other.

Rock

33 0 28m

46 m

11.5 Applications of right and

non-right-angled trigonometry

Trigonometry is a tool for solving
problems that involve triangles. There

are many examples of, and uses

for, trigonometry in our lives—from
astronomy and navigation to surveying

and sports. Sometimes we need to know
measurements that we cannot find

directly, such as the height of a building
or a distance to a ship at sea. Architects,
scientists and engineers use trigonometry
to determine all sorts of things—from the
sizes and angles of parts used in engines
to the distances to stars.

In order to make use of trigonometry, you have to apply the
appropriate formula for the situation that you are working with. Here

is a guide:

492

11.5

Use the Pythagorean theorem in a right-angled triangle when you know the lengths
oftwo sides of the triangle, and you want to find the length ofthe third side.

Use SOH-CAH-TOA in a right-angled triangle when you have two sides or one side
and one angle.
Use the sine rule when you have the measures of an angle and its opposite side,
along with one more piece of information.

Use the cosine rule when you know the measures oftwo sides and the included
angle, orwhen you know the measures of all three sides.

Example 12 s

Jon and Jan have two gardens. Jon has garden PQR 13m

and Jan has PRS.

a Find the length of the fence, PR, between their 7m

gardens. 300
b Find the area of Jon's garden.
12m
c Find PÉR in Jan's garden.

a q2 = p2+ t-2—2 pr cosQ You have two sides, p and r and the
q2 = 122 + 72-2 x 12 x 7 x cos300
-47.507 ... included angle, Q.
q = 6.89m
Select the cosine rule.

A = —pr sin Q Use — ab sin C to find the area.

sin 30 0 = 21 m2 2

2 You have side r and angle R opposite side s.

sin S sin R Select the sine rule.

c

sin S sin 100

6.89 13

6.89 sin 100

sin S =

13

31.50

Exercise ill Parallax angle Earth's position
in June
1 Calculate the area of the triangle with sides "Nearby" qar
Sun
measuring 12 cm, 14 cm and 20 cm. p
Earth's position
2 The radius of the Earth's orbit around the
Sun is approximately 150 million km. in December

When the parallax angle to a nearby 493

star is 70, find the distance from the
Earth to the star.

Earth's orbit

11 RELATIONSHIPS IN SPACE: GEOMETRY AND TRIGONOMETRY IN 20 AND 3D

3 a Find the length of [PR]. A6 toy bicycle frame is constructed from five
b Find the length of [QRI.
C Calculate the area of triangle PQS. plastic pieces.

a Calculate the length of [ACI.

b Calculate the size of BÄC.
c Write down the size of ACID.

d Calculate the length of [AD].

14 m e Calculate the area of the quadrilateral

ABCD.

500 550 5 cm c
R 950
11m s

4 A flagpole is supported by three wires BA, 320 480
D
BC and BD, as shown in this diagram.
TWO security cameras are positioned on the
a Calculate ADB.
b Find the area of triangle ABD. mceiling of a gym, 10 apart.

c Find the angle of elevation of the top of One camera has an angle of depression of
the tower from point C. 500 to a point on the floor, and the other
camera has an angle of depression of 600
d Show that triangle ABC is not right to the same point. Calculate the height, h

angled. metres, of the gym.

28m 600 500
13m

12m

c

S A dockyard crane ABC, is used to unload HINT
cargo from ships. AB = 22.5 m, AC = 48m
and angle ABC = 460. North

a Find AOB. In bearing problems North
b Find the length of the lifting arm, BC.
we often encounter
c
two parallel north lines.

The anglesÄ and are

supplementary angles,
they add upto 1800.

For example, when angle

Ä is 600, angle is 1200.

A8 Porpor flies from to B on a bearing of

0670 for a distance of 80km and from there

to C on a bearing of 1230.
a Find ABC.

b Calculate the distance for Porpor to fly

from C back to A.

494

11.5

c Find the bearing that Porpor must fly on 11 A right-angled ABCD with height OV = 20 cm

to return to A from C. has a square base of side I Ocm. Find

N 80m B 1230 a the length of the slant edge

670 120m b the angle between a sloping face and the

base

C the angle between two sloping faces.

c

9 A ship sails from harbour H on a bearing of c

km0840 for 340 until it reaches point P. It 20 cm

then sails on a bearing of 2100 for 160km

until it reaches point Q.

a Calculate the distance between point Q 10cm

and the harbour. 12 a Calculate the value of cos ABC in this

b On what bearing must the ship sail to triangle.
return directly to the harbour from Q?

840 340 2100 6
H 160 km 8

10 A ship leaves Shanghai on a bearing of c
12

b Without actually calculating the size of
the angle you will be able to say that

ABC is obtuse. Using your answer in part
a, explain why you are able to do this.

km0300. It sails a distance of 500 to Jeju.

At B, the ship changes direction to a
bearing of 1000. It sails a distance of

320 km to reach Nagasaki.

Calculate the bearing and distance for the
ship to return from Nagasaki to Shanghai.

Developing your toolkit
Now do the Modelling and investigation activity on page 504.

495

11 RELATIONSHIPS IN SPACE: GEOMETRY AND TRIGONOMETRY IN 20 AND 3D

Chapter summary

• The midpoint of (xvY1,z1) and (xyY'2,z2) is
2 22

-(• The distance (d) from (xvY1,z1) and (xyY'2,z2) is d — 2

• A pyramid has a base that is a polygon and three or more triangular faces that meet at a point
called the apex. In a right-angled pyramid, the apex is vertically above the centre of the base.

triangular face

height / I

base

Square-based pyramid Tetrahedron Hexa9nal-based pyramid

• A sphere is defined as the set of all points in three-dimensional space that are equidistant from a

central point. Half of a sphere is called a hemisphere.

height

slant
height

radius

radius

radius

Cone Sphere Hemisphere

Surface area

• The surface area ofa pyramid is the sum of the areas ofall of its faces.

• The surface area ofa cone is the sum of the areas of the circular base and the curved side.

Shape Surface area Volume
Pyramid
The sum of all of the faces. V = —(basearea x height)
Cone
x2 + 2xl for a squared-based pyramid where x is V = —xr2h

the side length ofthe base and lis the slant height. 3

7tr2 + rrl, where ris the base radius and lis

the slant height.

Sphere V = —mr3

3

O

496

11

O
Right-angled triangle trigonometry
• In a right-angled triangle:
— —sin 0 = —,cos0 = and tano =

AOwhere is the opposite side to angle 0, is the adjacent side and His the hypotenuse.

There are two special right-angled triangles that will give you exact values for the trigonometric ratios.
The first one is an isosceles right-angled triangle and the second is a 30-60-90 right-angled triangle
where the side opposite the 300 angle is half of the length of the hypotenuse.

45 600
2
1
1

450
1

sin 300 450 600
cos 9 2
1 2 2
tan
2 1 1

2 2

3

Angles of elevation and depression angle of
elevation
When looking up, the angle of elevation is the angle up from
horizontal
the horizontal to the line from the object to the person's eye.
angle of
• When looking down, the angle ofdepression is the angle depression
down from the horizontal to the line from the object to the

person's eye.

Bearings w

• A bearing is used to indicate the direction of an object 2400
from a given point. Three-figure bearings are measured 600
clockwise from North, and can be written as three figures,
such as 1200, 05? 0, or 3140. Compass bearings are s
measured from either North or from South, and can be
written such as N200E, N550W, S360E, or S6?0W, where the O
angle between the compass directions is less than 900.
Continued on next page
The bearing of A in the diagram may be written as 2400 or

as S600W.

49?

11 RELATIONSHIPS IN SPACE: GEOMETRY AND TRIGONOMETRY IN 20 AND 3D

O
Non-right-angled triangles

The area ofa triangle can be found using A —ab sin C where a and b are two sides and Cis their

included angle.

• sin A sin B sin C
. It is usual to only use two of these fractions.
The sine rule is b
c
a

• sin A = sin B when A + B = 1800.

AABC• The cosine rule to find a side in is:

a2 = b2 + ca — 2bccosA

or b2 = a2+c2—2accosB or +b2—2abcosC

AABCThe cosine rule to find an angle in is:

cosA = —a 2

2bc

2 +c2 —b2

or COSB =

2ac

or cosc =

2ab

Use the Pythagorean theorem in a right-angled triangle when you know the lengths oftwo sides, and
you want to find the length ofthe third side.

Use SOH-CAH-TOA in a right-angled triangle when you have two pieces of information.
Use the sine rule when you have the measures of an angle and its opposite side, along with one
more piece of information.
Use the cosine rule when you know the measures of two sides and the included angle, or when
you know the measures of all three sides.

Developing inquiry skills

Chicky is a glass cleaner halfway up of
one ofthe edges ofthe Louvre pyramid.
How far is it to the opposite comer of the
base?

498

11

Chapter review Click here for a mixed
review exercise
1 Find the volume and the surface area of
a square-based pyramid with a base side a Calculate the volume of the cylinder in

length of 8m and a height of 3 m. cm3, correct to 2 decimal places.

13m The cylinder is used for storing tennis
balls. Each ball has a radius of 3.25 cm.
8m
2 Find the volume and the total surface b Calculate how many balls Jamie can fit in
the cylinder when it is filled to the top.
area of a cone with radius 6cm and height
c Jamie fills the cylinder with the number
8 cm. Write your answer as an exact value of balls found in part b and puts the lid
(this means in terms of a). on. Calculate the volume of air inside
the cylinder in the spaces between the
8 cm
tennis balls.
6 cm
d Convert your answer to part c into cubic
3 A sphere has a volume of —n32 m3. Find the
metres.
exact value of its surface area.
A6 metal fuel tank is made of a cylinder
Find the volume of a cone with height of length 8.5m and diameter 3m with a

10 cm and base radius 4 cm. hemisphere welded to each end.
a Find the volume of the fuel tank.
b A truncated cone is formed when the b Find the surface area of the fuel tank.

top of the cone is removed with a 3 cm

horizontal cut, leaving a height of 6cm 8.5 cm

and top radius of 2 cm. Find the volume ABCDV is a solid stone pyramid. The base of
of the truncated cone
the pyramid is a square of with side length
6 cm. The vertical height is 3 cm. The vertex

OV is directly above the centre of the base.

11

10 cm 2 cm c
6 cm
a Calculate the volume of the pyramid.
4 cm b The stone weighs 12 grams per cm3.

S Jamie has a circular cylinder with a lid. The Calculate the weight of the pyramid.

cylinder has a height of 39 cm and diameter c Show that the length of the sloping
65 mm.
edge [VCI of the pyramid is 5.20 cm.
d Find the angle at the vertex, BVC.
e Calculate the total surface area of the

pyramid.

499

11 RELATIONSHIPS IN SPACE: GEOMETRY AND TRIGONOMETRY IN 20 AND 3D

A square-based pyramid, with the 13 The length of the base of a cuboid is three
times the width x, and its height is h cm.
Acoordinates of (l, 0, 3), B (l, 5, 3) and Its total surface area, A, is 600 cm2.
6,E (7,
10) is shown. Find: h
x
a the length AB

b the midpoint of [ABI

c the length AE.

a Show that A = 6x2 + 8xh.
c b When A = 600, find an expression for

h, in terms of x.

C Show that the volume •IS (100-x
2)

mFrom the top of a vertical cliff 70 high,

9 Find the area of the triangle: an observer notices a yacht at sea. The

4 cm angle of depression to the yacht is 450. The
300
yacht sails directly away from the cliff, and
6 cm
after five minutes the angle of depression

Howis 100. fast does the yacht sail in

km In-I ?

10 80 metal, spherical cannon balls, each of A15 radio antenna, [ABI, 80 metres tall, is
diameter 16 cm, were salvaged from a
on the roof of a communications building.
pirate ship.
c
a Calculate the total volume of all 80
cannon balls to the nearest whole AThe angle of depression from to a point
number.
C on the horizontal ground is 320.
The cannon balls are to be melted The angle of elevation of the top of the
building from C is 150.
down to make a cone.
Find the height of the building.
The base radius of the cone is 40 cm.
16 a Find the smallest angle of a triangle
b Calculate the height of the cone.
with side lengths 6, 7 and 8 cm.
11 Your teacher asks you to measure the height b Find the area of the triangle.
of a flagpole. You stand at a point 10 metres
from the base and measure the angle of 17 The diagram shows a triangular region
formed by a river bank, a wall [ABI of
elevation to the top of the flagpole as 600.
length 34m and a fence [BCI. The angle
Howa tall is the flagpole?
between the wall and the river bank is is
Your friend stands directly behind you 250. The end of the fence, point C, can be
and measures the angle of elevation to positioned anywhere along the river bank.

the top of the flagpole as 300.

b What is the distance between you and

your friend?

12 Find the surface area of a tetrahedron with
all faces equilateral triangles of side 6 cm.

500

11

c river bank

250 70 km 1500
100km
34 m

a Find the length of the fence [BCI, c

mwhen point C is 15 from A. a Find Aic.

The farmer has another, longer fence. b Hence find the bearing of mast A from
It is possible for him to enclose two
different triangular regions with this mast C.
fence. He places the fence so that
Afic = 950. C Find the distance from mast B to mast C.

Ab Find the distance from to C. 20 The Leaning Tower of Pisa, [PQI, leans at
c Find the area of the region ABC with 50 away from the vertical.

the fence in this position. A mGalli stands at point on flat ground 119

d To form the second region, he moves away from Q in the direction of the lean.
the fencing so that point C is closer to
He measures the angle of elevation to the
point A. Find the new distance from top of the tower P to be 26.50.

18 This plan shows a plot of land PQRS p

crossed by a path QS. : 900
95 26.50
ps
a Find the measure of angle QPA.
850 b Find the length, PQ, of the Leaning

12m Tower of Pisa.

30m 420 C Calculate the vertical height PG of the

a Find QS. top of the tower.
b Find the area of triangle QRS.
Exam-style questions
The area of triangle PQS is half the area
A21 P2: squared-based pyramid has slant height
of triangle QRS.
c Find the possible values of 0. 7 centimetres. The edges of the base are 5
d Given that Ois obtuse, find PS. centimetres long and the apex is located
vertically above the centre of its base.

19 The diagram below, which is not drawn 7 cm

to scale, represents the positions of three 5 cm

mobile phone masts. 5 cm

The bearing of mast B from mast A is 1000 a Find the total surface area of the
and they are 70 km apart.
pyramid. (2 marks)
The bearing of mast C from mast B is

1500.

Masts A and C are 100 km apart.

b Calculate the volume of the pyramid.

(4 marks)

501

11 RELATIONSHIPS IN SPACE: GEOMETRY AND TRIGONOMETRY IN 20 AND 3D

A22 P2: newly built tower is in the shape of a a Calculate the spinner's surface area.

cuboid with a square base. The roof of the (4 marks)
tower is in the shape of a square-based
b If the spinner is placed in a cylinder-
light pyramid. shaped container with height 10.1

The diagram shows the tower and its cm and diameter 6.1 cm, determine

roof with dimensions OE = OF = OG the percentage of the volume of the
= OH = 10m, AB = BC = CD = AD = 6 container occupied by the spinner.

m and AE = BF = CG = DH = 42 m. (2 marks)

10m 24 P2: Consider a trapezium ABCD with AB =
CD and BC parallel to AD. Given that AB
= 12 cm, BC = 13 cm, and CD = 22 cm,

a show that the trapezium has height

p 12 centimetres.

(3 marks)

42 m b Hence, find the area of the

trapezium. (2 marks)

C Find the size of the angle C.

A 6m B (3 marks)

a Calculate the shortest distance from d Calculate the length of the diagonal
AC. (3 marks)
O to EE (2 marks)
A A25 Pl: ship leaves a port located at point
b Hence, find the total surface area of
on a bearing of 0300. It sails a distance
the four triangular sections of the
of 20 km to a point B where it changes
roof. (2 marks)
direction to a bearing of 750. Then it sails
C Calculate the height of the tower
km25 to reach a port located at a point C.
from the base to O. (2 marks) a Find the distance between A and C.

d Determine the size of the angle

between OE and EF. (2 marks) (4 marks)

A bird nest is perched at a point P on the b Find the bearing of the point C with
Aedge, CG, of the tower. person at the
respect to A. (3 marks)

point B, outside the building, measures 26 P2: The diagram shows a cuboid ABCDEFGH.
the angle of elevation to point P to be 600.
The vertex F is located at the origin of a
e Find the height of the nest from plane and the vertex C has coordinates
(8, 10, 6). The faces of the cuboid are
the base of the tower. (2 marks)
parallel to the coordinate planes.
A23 P2: spinner is made of two equal cones
z
with heights 5 cm and radii 3 cm, as

illustrated in the diagram.

5 cml D Gy
3 cni c

5 cm xH

a Show that the length of the
diagonal of the cuboid is 1005 cm.

(2 marks)
b Determine the coordinates of the

Mmidpoint of the segment AD.

(2 marks)

502

11

C Find the distance between the Calculate:

points F and M. (2 marks) a the volume of the container

d Find the perimeter of the triangle (4 marks)

CFM, giving your answer exactly. b the total surface area of the

(3 marks) container. (3 marks)

e Show that cosine of the angle AMF 29 P2: The diagram shows a water tower with
height 30 meters and width 3 meters.
10 (3 marks) The tower stands on a horizontal
is equal to
platform.
10

2? P 1: The diagram below shows a

quadrilateral ABCD such that

10 cm, AD cm, BC=13 cm,
BCD = 450 and BAD = 300.

c

450

13 cm 3 cm

5 cm B From a point A on the ground the
300
angle of elevation to the top of the
10 cm tower is 320.

a Find the exact area of the triangle a Calculate the distance x, giving your
answer correct to the nearest metre.
ABD. (2 marks)

(3 marks)

(4 marks) b Hence, determine the distance y

C Find the exact value of sin CDB. between A and B. (2 marks)

(3 marks) AC Find the angle of depression of

d Hence, explain why there are two from B.

possible values for the size of the (2 marks)
angle CBD, stating the relationship
30 P2: The diagram shows an obtuse triangle
between the two possible values.
ABC with AB = 57 cm, AC = 48 cm and
(2 marks)
BÄC=1170.
28 P2: The figure shows a container in the
shape of a circular cylinder topped by a 48 m
hemisphere. The radii of both cylinder
and hemisphere is 3 centimetres and the 1170
height of the cylinder is 7 centimetres.

57 cm

Calculate:

a the length BC (3 marks)
b the area of triangle ABC (2 marks)

c the size of the angle ABC. (3 marks)

7 cm

503

Three squares Approaches to learning: Research, Critical

Thinking

Exploration criteria: Personal engagement
(C), Use of mathematics (E)

1B topic: Proof, Geometry, Trigonometry

The problem
Three identical squares with length of 1 are adjacent to one another. A line

is connected from one corner of the first square to the opposite corner of
the same square, another to the opposite corner of the second square and
another to the opposite corner of the third square:

cD

Find the sum of the three angles a, /3 and (P.

Exploring the problem

Look at the diagram

What do you think the answer may be?

Use a protractor if that helps.

How did you come to this conjecture?

Is it convincing?

This is not an accepted mathematical truth. It is a conjecture, based on

observation.

You now have the conjecture a + [3 + (P = 900 to be proved

mathematically.

Direct proof
What is the value of a?
Given that a + [3 + (P = 900, what does this tell you about a and [3 + (P?

What are the lengths of the three hypotenuses of AABC, AABD and
AABE?
Hence explain how you know that AACD and AACE are similar.
What can you therefore conclude about CAD and CÉA?
Hence determine why AOB = CAD + ADC and conclude the proof.

504

11

Proof using an auxiliary line

An additional diagonal line, CF, is drawn in the second square and the
intersection point between CF and AE is labelled G:

cD
Explain why BAC = a.
Explain why EÄF = (P.
If you show that GAC = D, how will this complete the proof?
Explain how you know that AGAC and AABD are similar.
Hence explain how you know that GAC = BBA = /3.

Hence complete the proof.

Proof using the cosine rule

The diagram is extended and the additional vertices of the large rectangle are

Xlabelled and Yand the angle is labelled 9:

x

Explain why XÉY = p.
Calculate the lengths of AE and AY.
Now calculate AÉY (0) using the cosine rule.
Hence explain how you know that [3 + = 450.

Hence complete the proof.

Extension

Research other proofs on the internet.
You could also try to produce a proof yourself.

You do not have to stop working when you have the proof.
What could you do next?
You could use the methods devised in the task in Chapter ?
on Spearman's Rank to rank the proofs and discuss the

results.

505

Periodic relationships:

trigonometric functions

Sine waves are the building blocks of sound. Concepts
In fact, all sounds can be created by combining
sine waves. Sound travels as a transverse wave. Equivalence
Musical instruments and electronic tools use Relationships
these waves to produce sounds of different
frequencies, volume and pitch. Microconcepts

Many phenomena that have a periodic, or Radians
Arcs and sectors
repeating, pattern, such as tides, sunlight Unit circle and the ambiguous case of the
sine rule
and Ferris wheels, may be modelled with Trigonometric identities and equations
Trigonometric periodic functions
trigonometric functions.
How can I model
my height on the

Ferris wheel?

Can I visualize
the amplitude,
pitch and
frequency of
musical notes?

What is the best time to surf?

What time is sunset in

Bali on my birthday?

506

The sounds we hear are caused by vibrations

that send pressure waves through the air. Our
ears respond to these pressure waves and send

information to the brain about the amplitude and

frequency of these waves. The brain then interprets

these signals as sound. 4.

A speaker usually consists of a paper cone attached

to an electromagnet. By sending an oscillating

electric current through the electromagnet, the Developing inquiry

paper cone can be made to oscillate back and forth.

When you make a speaker cone oscillate 440 times
Aper second, it will sound like a pure note.

We need to describe oscillations 1

that occur many times per 0.5 skills

second. The graph of a sine Research how Bach and other

function that oscillates through composers used mathematics in their

one cycle in a second looks like

this: musical compositions.

-0.5

Research graphs of sounds You might start by watching the Ted
waves for different notes. -1 Talk on music and maths, available on

o 0.2 0.4 0.6 0.8 1 YouTube.

What is the same about all Time (s) Think about the questions in this

these graphs? opening problem and answer any you

What is different? can. As you work through the chapter,

What is the relationship between a graph of a you will gain mathematical knowledge
sine function thatpscillateythrough one cycle and skills that will help you to answer
them all.
pe€second above) and the graph o sine
funcüon ha oscillates through tw cles per

Before you start Skills check Click here for help
with this skills check
You should know how to:

1 Find the exact values of certain trigonometric 1 Find the exact value of each.
a sin 450
ratios. b tan 600
C cos 300
eg Find the exact value of sin 300.

sin 300 = 0.5

2 Work with the graphing functions of your GDC. 2 Use the graphing functions of your

eg Use the graphing functions of your GDC to find the x-intercepts of the

GDC to find the x-intercepts of the graph of each function.

graph of f(x) = x2 — 3x— 6. a f(x) — 2x2 + 1

x--1.37, 4.37. b f(x) = ex—2

eg Use the graphing functions of your 3 Use the graphing functions of your

GDC to solve the equation, GDC to solve each equation.

31nx = 2 —x2 a eX=2x2-5

x = 1.20 b Inx=3—x

12 PERIODIC RELATIONSHIPS: TRIGONOMETRIC FUNCTIONS

12.1 Radian measure, arcs, sectors

and segments

Investigation 1

For this investigation you should be in a group of two or three.
You will need: compass, protractor, string scissors, poster paper, pencil and glue.

1 At one end of the paper, construct a circle on a set of axes with radius 10 cm.
2 Use a protractor to measure and mark every 100 around the circle.

3 Tothe right ofthe circle draw an x-axis about 65 cm long and draw a y-axis
about 20 cm tall.

4 Place the string around the circle with the end at zero. Mark the 100 marks

onto the string. Then put the string on the x-axis of the right-hand set of
axes and transfer the marks labelling every 100.
S Measure the vertical distance from the 100 mark to the x-axis. At your
mark of 100 on the other set of axes, draw a line segment the same height
as the one you just measured.

1

10 0 10 0

-1

6 NON repeat for 200, 300 and so on, until you have gone completely around the

circle to 3600.
? Take your string and glue it to your poster from zero degrees, along the

ends of the line segments to form a smooth curve.

This is the sine curve.

8 Research periodic functions. Is the sine function periodic? Why?
9 Factual What is a periodic function?
I Factual What shape is the graph of a periodic function?

11 Use your calculator to graph sin x. Does it look like your string graph?

12 Conce tual How does the shape of a periodic function show that there
will always be multiple xvalues that give the same values ofy?

13 Write an explanation about why sin 600 = sin 1200.
14 Conce tual How do we extend trigonometric ratios beyond right-angled

triangles?

508

12.1

O 15 Discuss your answer to the following questions with your group and then

share your ideas with your class.

a What is the radius of the circle in string units?
b What is the circumference in string units?
c What happens after 3600?
d What are the zeros of your sine function?
e Where is the highest point above the x-axis? At what angle is this?
f Where is the lowest point below the x-axis? At what angle is this?
g How would you do this investigation differently for a cosine curve?

You already know how to use degrees when measuring angles, but

most science and engineering applications use radians. Radians are an

alternative way of measuring the size of an angle.

Investigation 2

You will need: string, a compass, a ruler, a protractor and a pencil.

1 Draw anx- and y-axis.
2 Use your compass to construct a circle with the origin as the centre.
3 Label the point where the circle cuts the positive x-axis 00, the point

where the circle cuts the positive y-axis 900 and keep going counter-
clockwise round the circle to label 1800, 2700 and 3600. You may notice
that 0 and 360 are in the same place.
4 Take your string and put it around the outside of your circle. Cut off any
excess string so that you have the exact circumference in string.
S Put one end of your string at the origin and mark the point where it touches
the edge of the circle. You should have marked the length of one radius on

the string.
6 Mark any point on the circle A. Put the end of the string here, draw it out

along the circumference of the circle and mark the length of one radius.
Mark this point B.

O? Mark the centre and draw segments [OA] and [0B].

8 Repeat this for several starting points on your circle.
9 Use your protractor to measure angle AC)B for each shape and average

those answers.

10 Share your results with the class by writing everybody's averages on the
board and averaging all of those values.

11 Use the Internet to search "What is a radian?"
12 About how many radians is a straight line? Can you think ofa mathematical

value or constant that is close to this number? We say that an angle which

forms a straight line measures IT radians.
13 Factual What is the relationship between radians and degrees?

14 Factual What are some uses of the radian measure?

15 Conce tual Why are radians dimensionless?

16 With a partner, discuss what you have discovered, what you now know
about radian measure and what you wonder.

509

12 PERIODIC RELATIONSHIPS: TRIGONOMETRIC FUNCTIONS

Radians

A radian is the measure of the angle with its vertex at the centre of the circle

Wobetween radii with endpoints that are one radius length apart on the

circumference of the circle.

1 radian

You might recall that the circumference of a circle is 2rrr, so that means HINT
Make sure you know
there are 211, or approximately 6.28 radians, in a full circle. This gives how to use your GDC
you the conversion: to work with both
degrees and radians.
2m radians = 3600 or I radian 57.30

From this you can deduce that:
rt radians = 1800

—radians = 90

2

—radians = 60 1 Y - sin

3

—radians = 45

4

—radians = 30 n n 3n

6

You can use radians in the place of degrees on -1

the sine curve from Investigation l.

Converting between degrees and radians

Degrees to radians: Radians to degrees:

Multiply the degree measure by Multiply the radian measure by

1800

180

The symbol "c" or the abbreviation "rad" are used for radians, but they

are often omitted, particularly when is present. The symbol for degrees

is never omitted.

Example 1 b Convert 56.50 to radians. —c Convert to degrees.

a Convert 200 to radians.

20m The 20 and the 180 cancel. Leave your answer
a 200 in terms of n.
The numerator and denominator do not
180 9 cancel. Give your answer to 3 s.f.

56.5m 0.986 rad The cancels. Do not forget the degree symbol.

56.50

180

4m 1800

c
33

510

12.1

Exercise 12A

1 Express each angle in radians. 3 Express each angle in radians, Discuss your
answerwith a friend
a 450 b 600 c 2700 d 3600 correct to 3 significant figures. then share your ideas
with your class.
e 180 f 2250 g 800 a 100 b 400 c 250 d 3000
e 1100 f 750 g 850 h 12.80 Who "invented" the
h 2000 i 1200 1350 i 37.50
j symbol for pi?
Write a sentence
2 Express each angle in degrees. 4 Express each angle in degrees,
stating who you think
a b c d 3rr correct to 3 significant figures.
6 10 6 it could have been.
a I rad b 2 rad c 0.63C Research to find the
4m correct answer.

d 1.41C e 1.55c

20 5 4
13m
g 0.36C h 1.28c i 0.01c
4 2.15 C
517

93

Sectors and segments

An arc is part of the circumference of a circle. Minor arc AB is the shortest

Adistance between two points and B on the circumference of a circle, major
arc AB is the longer distance around the circumference from A to B. Unless

otherwise indicated in a question, we assume arc AB refers to the minor arc.

sector is the region enclosed by two radii and an arc. Unless otherwise

AOCindicated in a question, we assume sector refers to the minor sector.

A chord is the line segment joining two points on the circumference of a circle.

segment is the region between a chord and an arc.

minor arc arc
minor sector
sector

c

major
sector

major arc segment

The length of arc, I, can be found by using the

circumference formula for a circle:

—xI — 2;rr = ro

2m

The area of a sector, A, can be found by using
the area formula for a circle:

0 21
2m 2

The length of an arc is given by 2
and the area of a sector by
where Ois measured in radians.

511

12 PERIODIC RELATIONSHIPS: TRIGONOMETRIC FUNCTIONS

Example 2 8 cm

Find: The exact answer would leave your arc

a the arc length and length in terms of m

b the area of the sector of this circle. You can also round to 3 s.f. to give 18.8 cm.
You can also round to 3 s.f. to give 75.4cm2.
—8 x = 6m cm

4

= 24m cm

4

Example 3

Find the shaded area.

6 cm

c Find the area of sector OACB. Leave your
answers as an exact value as you will be
= 12m
using this again.
f)
Find the area of triangle OAB.
— x6x6 x sin2—m = 18 sin—
Area of the sector — area of the triangle
2 33

Shaded area = 12;r —18 sin2—m = 22.1 cm2

3

Exercise 12B (9)

1 Find: iv
a the arc length 14n
b the area of the sector in each circle.
3m
i ii
5n 15cm
3n 12m
14 cm —2 The sector formed by a central angle of
12
has an area of 37t cm2. Find the radius of
the circle.

512

12.2

A3 sector has a radius of 12m and area of 6 A complete turn has 360 degrees, and

36Tt m2. Find: each degree is divided into 60 parts called
a the central angle which forms the sector minutes.
b the perimeter of the sector.
When the central angle with its vertex at
4 Find the shaded area when e = 1.5 rad.
the centre of the earth has a measure of
10 I minute, the arc on the surface of the
earth that is formed by this angle (known
o 10 as the great-circle distance) has a measure

of I nautical mile.

The radius of the earth is approximately

6371 km. Find how many km there are in

I nautical mile.

1 Nautical mile North Pole

S The pendulum on a large Equator
grandfather clock swings from
one side to the other once

every second. The length of

mthe pendulum is 4 and the

angle through which it swings

is —. Find the total distance

12

the tip of the pendulum
travels in one minute.

Developing inquiry skills

How would you describe the shape of the function in the opening problem?

Is this periodic function? Explain your reasoning.

12.2 Trigonometric ratios in the unit

circle

The unit circle is a circle with its centre at the origin and of radius 1 unit.

(0,1)

(-1,0) 1
(1,0)

x

(0,-1)

513

12 PERIODIC RELATIONSHIPS: TRIGONOMETRIC FUNCTIONS

Angles in standard position in the unit circle are measured starting
on the positive x-axis and turning anticlockwise (positive angles) or

clockwise (negative angles).

x x
(1,0) (1,0)

The four quadrants 900

The coordinate axes divide the Second quadrant First quadrant
plane into four quadrants, labelled 1800 Fourth quadrant
first, second, third and fourth,
as shown. Angles in the second Third quadrant
quadrant, for example, lie between
900 and 1800. 2700

Investigation 3

Work in small groups. This investigation uses degrees, but it works equally
well if you want to convert the degrees into radians.

1 Draw a unit circle on graph paper.

2 Now draw point A on your circle with 0.5

=0 300 and draw a line vertically to the

x-axis to make point B.

3 Use trigonometry to find OB. This is the 0
x-coordinate of point A.
-0.5
4 Use trigonometry to find AB. This is the

y-coordinate of A.

S Write down the coordinates of A.

Notice that the coordinates of A at any angle 9 are (cos 0, sin 9).

6 Repeat your drawing of A and find its coordinates when
600, 1200, 1500, 2100, 2400, -300, -600 and -1200.

? Explain to your group what you notice about:

sin 300 and sin 1500 cos 1200 and cos —1200

sin 600 and sin 1200 tan 300 and tan 2 100

cos 300 and cos —300 tan 600 and tan 2400

cos 600 and cos —600

The sine graph from investigation 1 can help you visualize these patterns.

8 In the graph ().5 is shown on the same axes withy= sin 0.

Can you see where the lines intersect? What do you think will happen

Owhen x is greater than 3600? What will happen when xis less than 00?

514

12.2

O

0.75
0.5

025

—300
-0

-0.

9 Conce tual Using symmetry, state the different relationships between

angles in different quadrants for each ofthe trigonometric values.

10 How are positive and negative angles of all sizes represented on a unit circle?

11 How can you find different angles with the same trigonometric values?

12 How can you find the exact values of the trigonometric ratios of

64320, —,—, and their multiples?

13 Did you notice any other patterns?

14 What happens when you add 3600 to any of those angles?

From your investigation, you should see that:

sin sin( 180 — 9)
cos + 9)

tan

Looking at your results, the x- and 9000r
y-coordinates of any point that
lies in each of the four quadrants, sino positive sino positive
cos0 negative cos0 positive
we can identify the sign of each of tano negative tano positive

the trigonometric ratios in a given 18000r n

quadrant. sino negative sino negative
cos0 negative cos0 positive
tano positive tano negative

3n
2700 or

To help remember the signs of the three Second quadrant First quadrant HINT
trigonometric ratios in the quadrants, you Sine
All The mnemonic
can see that all of the ratios are positive in Tan
the first quadrant, then only one of the ratios Third quadrant Cosine All Students
is positive in the other three quadrants. Fourth quadrant Take Calculus

Remember that the standard position of an angle is the position of is a good way to
remember this.
an angle with its vertex at the origin of a unit circle and one side fixed
on the positive x-axis. This side is called the initial side of the angle.

515

12 PERIODIC RELATIONSHIPS: TRIGONOMETRIC FUNCTIONS

The other side of the angle, called the terminal side, will intersect the

circle at a point.

Angles with terminal sides that intersect the 1 1200

circle at points with with the same y value have 2'

equal sines. For example: 1

sin 600 = sin 1200 Discuss
your answer with a
sin 2400 = sin 3000 partner then share
your ideas with your
Angles with terminal sides that intersect the 1
circle at points with the same x value have class.
equal cosines. For example: 2400
What facts do you
cos 600 = cos 3000 know about the unit
cos 1200 = cos 2400
circle?
Angles with terminal sides that are directly opposite each other on a
line drawn through the origin have the same tangents: What can the unit
tan 600 = tan 2400 circle help you to

tan 1200 = tan 3000 Research why we use

In the unit circle, an angle can be used to form a right triangle with a radians instead of
hypotenuse of length one and two shorter sides of sin 0 and cos 0. degrees.

The Pythagorean identity gives you: (cos9, sine)
sin2Ø+ cos20= I 1

HINT

Notice that (sin is written sin2 0. cos 0

In the same unit circle diagram:

tan 0 = opposite side sin 0
cos0
adjacent side

sin2 0+ cos2 9

tan O = sin O

cosO

Example 4

1 Find an obtuse angle that has the same sine as the given acute angle.

a 700 b
4

2 Find another angle less than 3600 or 2m that has the same cosine as the given angle.

2m

a 500 b

3

1 a sin 700 = sin (180 - 70) 0 = sin 1100 TWO angles have the same sine if the angles
add up to 1800 or rt radians.
—b sin—= sin m —— = sin
TWO angles have the same cosine if the
4 44 angles add up to 3600 or 2m radians.

2 a cos 500 = cos(360 - 50)0 = cos 3100

b cos—=cos 217 — = cos—

3 33

516

12.3

Example 5

Without finding 0, find cos0, when sin 0 = 3 and is acute.

—

5

sin2 9 + cos2 16 Use the Pythagorean identity.
25 Use the positive value, as the angle is acute.
2

3

+ cos2 0 = I

5

2

cos2 e = 1— 3

5

4
cosO =

5

Exercise 12C

1 Find the sign of: 3 Find an angle less than 3600 or 27t
a cos 1300 b sin 3200 that has the same cosine as:
c tan 2250
a 400 b 1100 c 3000 d 5000

2 Find an obtuse angle that has the same h

sine as: 8 10 2 4

a 360 b 500 c 850 4 Solve these equations for 0 2m

d 4600 —a sino = —1 b cosO = b tano =

2m 3 5 22

7 h —S a Find sin 9, when cosO = 8 and 9is acute.

3 b Find tan 0.

Developing inquiry skills TOK

In the graph on the opening page, what do you think will happen after one
second? Can you describe the next few seconds? What do you think could
have happened before zero seconds?

12.3 Trigonometric identities and It is fascinating to see
how mathematics is
equations passed from nation to
nation, irrespective of
Solving trigonometric equations race or religion.
During this course, you have learned to solve many types of
Research the etymology of
equations, such as quadratic, exponential and logarithmic. the word "sine" and write a

Now we will look at trigonometric equations. paragraph explaining when
it was first used and why.
When solving trigonometric equations, it is important to know whether
Share your ideas with
your answer should be exact, in radians or degrees, within what a partner and compare
interval, or domain. "Exact" will often mean radians in terms of n. answers.

12 PERIODIC RELATIONSHIPS: TRIGONOMETRIC FUNCTIONS

Investigation 4

1

1 Solve sin 0=—using graphing technology.

2

2 Now find sin-10.5 on your calculator.

—Note that your calculator will give you 300 (or radians).
6

3 Graph sin x and y = 0.5 on the same set of axes.

4 Notice that the graph has more than one intersection, but your calculator
only gives one value. Show this answer on a unit circle. This is called the

primary value.

5 Now show 3900 on your unit circle.

6 Show 7500, 1 1 100, -3300 and -6900. What do you notice?

Enter the sine of each of these angles on your calculator.

8 Now find sin 7500 and then sin-I of the answer. What do you notice?

1

9 Discuss with a partner how to write all of the solutions to Sino =

2

Show this angle on a unit circle. This is called a secondary value.

1

10 Go back to your unit circle and find anothervalue for sin O — —,

2

11 Now show 5100 on your unit circle.

12 Show 8700, 12300, -2100 and -5700. What do you notice?

13 Enter the sine of each of these angles on your calculator.

14 Now find sin 5100 and then sin-I of the answer. What do you notice?

15 Factual How can you use these patterns to solve equations?

Conce tual How is trigonometry used to find all possible angles for
unknown values? How is trigonometry used to find unknown values?

1

1? Discuss with a friend how to improve your solution to sin O =

2
18 Share your final results with the class.

You will know from chapter I I that one solution to sin 0 —- — is 30 (or

and from the unit circle you saw that angles with

terminal sides that intersect the circle at points with

the same y-value have equal sines, and angles with 3) 2'
the same x values have equal cosines.

30

You also saw the identity sin sin(180 — 9), which

can help you to find the second angle.

Now you have —300 and 1 500, or— and
6

When you do a whole rotation, after each angle you

can add or subtract 3600 or 2n and so on. This will give you an infinite

number of solutions and so you are usually asked to solve an equation

within a specific domain.

— —o = or is called the primary value and 0 = 1500 or is the
66

secondary value and you can add or subtract as many complete turns

to both as you wish to find further angles in the given domain.

518

12.3

Example 6 2m 2 Solve —2 cos2x— sinx + I = 0 0 s x 4Tt.

1 Solve 2 cos 0 — I = O for O S Rearrange the equation.

1 2cose-1=o TWO angles have the same cosine when the

2 cose= I angles add up to 27t radians.
cos 0 = 1
Answers for 0 2m
2

0 = cos-l — —

2 3' 3

2 —2 cos2x — sinx + I From the Pythagorean identity,
—2(1 — sin2x) — sinx+ I cos2 0 = I — Sin2Ø

—2 + 2 sin2x — sin x + I Form and solve by factorization, as you
2 sin2x — sin x I would with a quadratic equation.

(2 sin x + l)(sinx— l)

2 sinx + I = 0 sin x I

sin x = ——1 sin x = I

2

x = sin-I 1 x = sin-II Find the inverse sine of the values.

2 Generate angles from the unit circle.

7m ll;r Add 2n to give angles up to 4m.

66 2

7m I lx 19m 23r

2' 2 6 6

Exercise 12D

1 Solve each equation for 0 3600, giving 4 Solve each equation for 0 2Tt, giving

your answers to I decimal place. your answers to 2 decimal places.

a cos 0=0.6 b sin 9=0.15 a 4cosx— 3sinx=0

C tan 0.2 d tan b 2 sin x + cosx=0

e cos e = —0.43 C tan2x— tan x —2=0

2 Solve each equation for 0 2m d 2 cos2x + sin x = I

a sin 0=0.82 b tan 9 = —0.94 5 Solve for in the given domain.

c cose=-o.94 d coso= 0.77 a cose= 0.3,

e sine--0.23 b tan 1.61,

3 Solve the equation 2 sin2 9+ 5 sin 3 for d 2 tan20+ 5 tan 3, —2rtS 0

6 Solve the equation 3 cosx= 5 sin x, for

0 S xS 3600, giving your answer to the

nearest degree.

519

12 PERIODIC RELATIONSHIPS: TRIGONOMETRIC FUNCTIONS

Example ? 2m.

—1 Solve sin 2x = for 0 < x
2

—1 sin 2x = Take the inverse sine.
2
4 Because we have 2x, and 0 x 2m, then
—2x = sin-I
we use 0 2x 4m When we divide these
2

3m 917

44 '4

values by 2 the answers we require will fall

in the domain 0 2m

3r 9m

8'8'8 8

Exercise 12E

1 Solve each equation for -1800 1800. 2 Solve each equation for 0 2m.

—a cos 2x = —b cos 3x = —a sin 30 = b cos 39-

2 2 2

C 2 cos3x I d 3 tan —+3 = 0 C sin — —d sire 20 -1=0

2 22 3

Double angle identities

Now we will look at the trigonometric formulas known as the

double angle formulas. They are called this because they involve
trigonometric ratios involving double angles, that is, sin 20

and cos 20. Trigonometric identities are equations relating
the trigonometric functions that are true for any value of the

variable. An equation is an equality that is true only for certain

values of the variable.

sino

You have already seen some trigonometric identities such as tano =

and sin2 9+ cos2 l. cosO

This unit circle shows the angles and —9. This makes angle BÄC = 20.

BD has length sin 0 and is the same length as CD, so BC = 2 sin 9.

Using the cosine rule in triangle ABC: B(cosO, sin 9)
1
BC2 = AB2 + AC2 - 2 20

BC2 = 12 + 12- 2 - 2 cos20

1

C(cos(-9), sin(-O))

520

12.3

Equating the two expressions for BC gives 4sin29= 2— 2 cos2 9.

Square both sides 2 cos 2 2 — 4sin2Ø.
Rearranging this equation cos 1 — 2 sin2Ø.
Divide both sides by 2

cos I — 2 sin20 is called the double angle identity for cosine.

There are two further identities for cos29 that you can work out using

the Pythagorean identity.

We know sin20 + cos2Ø= l, so sin2 I — cos2

Using substitution, we would have cos2Ø= I —2(1 — cos29).

Rearranging this equation gives us cos2ø= 2 cos2 0 — l.

You can substitute sin2Ø + cos29= I into this equation to get:

cos 2 cos2Ø— (sin20 + cos2 0)
cos 20 = cos29 — sin2 9.

You can also use the Pythagorean identity to derive an identity for sin 29.

You know that sin2A + cos2A = 1

When you let A = 29, you get sin220+ cos2 1

substitute cos20= I — 2 sin2 0 sin22Ø+ (l 1

Rearrange sin22 0 I — (l — 2 sin 29)2

Expand the bracket and simplify sin220= 4sin20— 4 sin4

Factorize the right-hand side sin220= 4sin2 — sin2 0)

cos2Ø= 1 — sin2 sin220= 4 sin2 (cos2 0)

Take the square root of both sides sin 29 = 2 sin 0 cos 9

This is the double angle identity for sine.

The double angle identities for sine and cosine are:
sin 2 sin ecos

cos I — 2sin2

cos 2 cos2

cos cos2 0— sin2 0.

Example 8

3 < find exact values for:

1 Given that Sino = — and 0 2

5

a cos b sin 29 C cos29 d tan 29. O

Continued on next page

521

12 PERIODIC RELATIONSHIPS: TRIGONOMETRIC FUNCTIONS

O sin2 0+ cos2 I Use the Pythagorean identity.
a
b 2 —Take the positive answer only, as 0 0

3 + cos2 0 2

5 Use the double angle formula for sine and

cos 0 1 2 9 16 substitute.
25 25
3 Choose any one of the three double angle

5 identities for cosine.

cos0 = —4 Use the tangent identity.

5

sin 29 = 2 sin ecos 0

sin 20 24
25

c cos29= I — 2sin2 0

2

cos20= 1 3 18 7
25 25
2

5

d tan 20 = sin 20 24 25 24

cos 20 25 7 7

Exercise 12F

1 Write each expression as a single S Solve each equation for 0 27t
trigonometric ratio.
giving your answers in terms of rt.

a 2 sin 5 cos5 b 2 sin— cos— a sin sin b cos29+ sin 0
c 2 sin4rtcos4rt
22 C sin20 = N6cosO d cos sin sin 29

d cos2 0.4 — sin2 0.4

e cos2Ø= cos

e 2 cos2(6)— I f sin2—
4
6 The expression 32 sin x cos x can be
2 Given that sino = —1 and is acute, evaluate expressed in the form a sin bx.

3 a Find the value of a and of b.

each expression.

a cose b sin 20 b Hence or otherwise, solve the equation

C cos 29 d tan 29 32 sin x cos x = 8, for 0 m

3 Given that cos O = ——1 and 0 is obtuse, find: Given that triangle ABC

2' has an area of IOcm2

a cose b sin 29 1

c cos 29 d tan 29 when sino =—, find x.

4

4 Given that Sino = ——1 and <—E, find:
2
8'

a sin 29 b COS 2

c tan 20 d sin 40 c
x

Developing inquiry skills

Notice that the function can have more than one intersection when you draw
horizontal lines on the opening graph. What would those lines tell you?
Where would there be only one point of intersection?
Is there anywhere that would be more than 2?

522

12.4

12.4 Trigonometric functions

The graph of a function provides a useful image

of its behaviour and allows you to see patterns. A

graph shows us a function's behaviour and makes

it easier to see the properties of a function. Data
analysis and problem solving frequently involve the
use of graphs and the understanding they provide to
allow for calculation and prediction. The graphs of
trigonometric functions have clearly visible

patterns.

The London Eye is a Ferris wheel with diameter
120 m. The wheel completes one turn every 30
minutes. After one cycle, it repeats its journey again
and again. This is an example of a periodic function.

A periodic function is a function that repeats its values

in regular intervals or periods. Tides, planetary orbits,

daylight hours, biorhythms, heart beats and musical
rhythms are some examples of phenomena that can be
modelled by periodic functions.

The London Eye repeats its revolution, or cycle, every

30 minutes, and so we say it has a period of

30 minutes.

The sine and cosine curves

You can see many similarities between the sine and cosine curves in

this diagram:

1 yec sx
0.5 x

-0.5 y-s nx
-1

The curves are the same shape.

The curves are the same height. Both functions have a maximum
value of 1 and a minimum value of —l.

Translating the cosine curve (the dotted line) —, horizontally to the

right, makes it identical to the sine curve.
Both functions complete one wavelength in 27t radians.

The functions are periodic, meaning they repeat the same cycle of

values.

523

12 PERIODIC RELATIONSHIPS: TRIGONOMETRIC FUNCTIONS

The characteristics of sine and cosine functions

This graph shows the function y = sin x. Y

The amplitude is half of the difference of the 1 UNOO

maximum and minimum values. That is, the y x

Avalue at point minus the y value at point B,

divided by 2.

-1

Investigation 5

1 Use technology to help you sketch the graph of each function for

a y=sinx b sin x c sin x

d Find the amplitude of each graph.

e How could you have found the amplitude by looking at the equation?

f Could the amplitude be negative? 2m
2 Sketch the graph of each function for 0

Y=—sinx b y=—2 sin x c y=—3 sin x

d Explain to a partner what you notice and explain the effect of the

negative sign.

3 Sketch the graph of each function for —27T<XS 2m

a Y= sin (—x) b y=2sin (—x) c y=3sin (—x)

d Explain to a partner what you notice and explain the effect of the

negative sign.

4 Use technology to help you sketch the graph ofeach function for() < x < 2m

a y=sinx b y=sin2x c sin 3x d y=0.5sinx

The period of a trigonometric function is the length of one full cycle along the
x-axis, ie the angle measure after which the graph begins to repeat.

S Explain to a partner what you notice.

6 Write an explanation of how the value of b iny= sin bxtransforms the

graph ofy = sin x

Factual What do the parameters represent inf(x) = a sin bx?

Conce tual How does changing the parameters affect the graphs of

trigonometric functions?

9 Conce tual How can you transform trigonometric graphs?

For the curvesy= a sin bxandy= a cos bx:

2m

—lal is the amplitude, and the period is

y=—sinx or y=—cosxis a reflection in the x-axis.
sin(—x) or y=cos(—x) is a reflection in the y-axis.

524

12.4

Example 9 The function has amplitude 3 and
Sketch the graph of y = 3 cos2x, 0 S x 2m
period = =
3
2 b

1

x
-1
-2
-3

Example 10 Y
0.6
Write down the period and amplitude, and 0.4
0.2
find the equation of this function.
0
x

Amplitude: 0.5 -0.4 () 5)
-0.6
Period: 217 -0.5.
3 max — min
Amplitude = 2
Y = 0.5 sin 3x
2

Period = 217 or 3 cycles in 2m radians.

3

The graph is the shape of a sine curve.

Note that you could also use a cosine function,
if you include a horizontal translation in your

equation.

Exercise 12G

1 Sketch the following graphs 2 Find the amplitude and period of each

for 0 S x 2m function.

a Y = 2 sin x b y = —3 cos x a Y = sin 3x b Y = 0.5 cos2x
c Y = 2 cos 2x
d y= 3 sin 4x c y = —4 cos3x d y = ——1 sin —x

23

525

12 PERIODIC RELATIONSHIPS: TRIGONOMETRIC FUNCTIONS

3 Write down the period and amplitude, and c x
2
find the equations of each graph.
1
a x

1 -1 uoox

0.5 -2
x
d
-0.5
-1 4 Let f (x) sin —x for 0 S x S 4.

b a Write down the amplitude of f.
3
2 b Find the period of f.

1 umua
x

-1

nuanun

-2

-3

Investigation 6

1 Use technology to help you sketch the graphs of the following functions

for —27T s x S 2m

a y=sinx b Y = sin (x —2)

c y=sin(x+2) d y=sin(x— l)

e How does the value ofciny= transform the graph of
y = sin x?

2 Sketch the graphs of the following functions for 0 2m

a y=sinx b y=2+sinx
c y=3+sinx d + sin x

e With a partner, discuss what you notice and explain the effect of the

constant.

3 Factual What do the parameters represent inf(x) = sin(x — c) + d?

a Conce tual How does changing the parameters affect the graphs of

trigonometric functions?

S How can you transform trigonometric graphs?

y=y=sin(x—c) or cos(x—c) translates the function cunits horizontally.

y = sin x + d or y = cos x + d translates the function d units vertically.

526

This leads you to the general sine (or cosine) function. 12.4

y=a sin +d Discuss
your answer with a
where lal is the amplitude. Amplitude = max — min partner then share your
ideas with your class.
—= 27t 2 • What do you see?
• What do you think?
Period • What do you

c is the horizontal shift and wonder?

max + min
d is the vertical shift. Vertical shift=

2

Example 11 Shape of a sine curve

Sketch the graph of y = 2 sin (3(x + a)) — Amplitude: 2 2m

1 b = 3; Period = 2m 3

nun x b

-1 Horizontal shift: —rt (or units left)

unqnuo Vertical shift: —1

-2
-3

Exercise 12H

1 Match each function with the correct graph:

i, ii, iii or iv.

a Y = 3sin 2x b y=—2cosx +1 iii

c y = 1 x +— d y = sin — x +2

—sin

2

i -2n 0 x

0.6 2n

iv

x

ii nananono
nonanana

x

x

12 PERIODIC RELATIONSHIPS: TRIGONOMETRIC FUNCTIONS

2 Find the equations of each function. 4 Solve the equation cos(x) — x2 = 0 by
sketching y = cosx and y = x2 on the same
a
axes and finding their intersection points.
5
4 5 Solve 2 sin x = x + 1 graphically.

3 1

2 6 Solve sin x =—, 0 S x 4rt graphically.

1 2

x 7 The equation ex = cos x has a solution
2n between —2 and —l. Find this solution.

b 8 Sketch y = sin x and y = cos x on the same

2 axes.

a Use your sketch to help you find the
value of cin sin x = cos(x — c).

1 b Copy and complete the sentence. The

graph of y = cos x may be translated

horizontally to the to

x become the graph of y = sin x.

c 2n 9 Let f (x) = sin(l + sin x). 6.

-1 x a Sketch the graph of y = f (x), for 0 < x
-2 b Write down the x-coordinates of all
-3 x
-4 minimum and maximum points of f.

d Give your answers correct to 4

significant figures.

10 Let f(x) = sin 2x and g(x) =sin(0.5x).

Find:

-1 a the minimum value of the function f

AVAVAV, b the period of the function g

3 Sketch the graph of each function for —C Solve f(x) =g(x) for 0 x
2
a y=2+ 3sin2x

b y = 0.5 sin x +—

3

c y = cos(x + n) — I
d y = 2 — 2 sin2x

e 2cos3(x + n) +1

The tangent curve
You will have noticed that the maximum value of the sine or cosine of
an angle is 1 and the minimum value is —l, but is this the case for the

tangent function?

528

12.4

Investigation ?

1 Copy and complete this table with the use of technology.

xo 20 40 60 80 90 100 120 140 160 180

tanx

x 200 220 240 260 280 300 320 340 360

tanx

y=2 Plot this data as points on a set of axes to give tanx.

3 Where do they cross the x-axis?
4 What happens at 900 and 2? 00?

S Connect your points to form smooth curves.

6 Is this a periodic function? Why?

Explain what you notice about the tangent function to a classmate.

8 How can you compare the graphs ofthe sine, cosine and tangent functions?

In radians, the tangent curve looks like this:

Y

x

There are some similarities to the sine and cosine curves. All three
curves are periodic functions. The transformations of horizontal and
vertical stretches, reflections and translations all follow the same
rules, though the period of the tangent function is 1800 or radians.

The tangent function does not have an amplitude since there are no

local extrema points.

Exercise 121

1 Sketch the graph of each function for 2 Use technology to help you solve each

equation.

a tan 20 b y = tan a tan 3, 2Tt

C y = —l + tan 0 2 b tane=x+ l, 2n

e y=tan 0—— d y = tan 9+ 2 C tan 9 = sin 9, —Tt

2 y=l+tan 0 —— d tan 5 — x2,

4

529

12 PERIODIC RELATIONSHIPS: TRIGONOMETRIC FUNCTIONS

Modelling periodic functions

Periodic occurrences happen in repeated and predicatable cycles, such

as lunar orbits and high tides. However, many real-life situations are

Wemore complicated than the simple rotation about a unit circle.

often need to transform the sine and cosine curves, so we can use

these functions to model an event.

One major difference with these graphs is that the original data might

not use angles as the independent variable. For example, our opening
problem with the London Eye will use time on the horizontal axis and
height on the vertical axis.

Investigation 8

The London Eye p;

The London Eye was the fourth largest Ferris wheel in
the world when it was built, with a diameter of 120 m.
One full rotation takes 30 minutes, and you board from

ma platform which is 2 above the ground. Can you

find your height above ground as a function of time in
minutes?

Factual What real-life situations can be modelled
with graphs of trigonometric functions?

Conce tual How can you solve real-life problems

using the graphs of trigonometric functions? p 11

3 As you ride, what stays the same? What changes? Ground 2m

Discuss your answers with a partner. p
11
4 How high above the ground would you be at the start of the ride (P )?

5 What is your maximum height (P6) above the ground during the ride?
6 How many minutes would it take you to get to point P ?

How many minutes would it take you to get top and what would be your height at that time?
8 How many minutes would it take you to get top and what would be your height at that time?
9 Copy and complete this table.

Position pp p pp p pp p pp

Time (mins) 12 3 56 8 9 10
Height (m)
5 20

2 35.?5 103.25

Conce tual How can you model periodic behaviour?

11 Use graph paper to plot these points and draw a periodic function to model yourjoumey on the London

OEye.

530

O 12 Find the amplitude, period and vertical shift of the curve. 12.4

13 Model the function as a cosine function. l!

14 Check your model by finding your height after 15 mins. This should be

the maximum height.

15 Factual What kinds of information can be modelled using periodic
graphs?

16 The best pictures of Big Ben are taken after 18 mins on the London
Eye. What will be the height at this time?

Factual What equations can be solved with trigonometric

functions?

Example 12

Ben is a fisherman with a boat in Rhyl harbour. This is part of a graph of the
depth of water at the mouth of the harbour.

8 omoaoomaoa
6 uuoaooxoououaooooxnouo

2

x

2 4 6 8 10 12 14 16 18 20 22 24

Hours after midnight (h)

a Find the amplitude of the function.
b Find the period of the function.

Model this as a cosine function.

c Find the horizontal shift.
d Find the vertical shift.

e Write down the equation of the function for the depth of the water in terms of

m and h.

f Calculate the depth of the water at 9:30am.

mBen's boat can only get in and out of the harbour when there is at least 4 of water at the

mouth of the harbour. Ben sleeps until

g Find the next time that he can go fishing.

Continued on next page

531

12 PERIODIC RELATIONSHIPS: TRIGONOMETRIC FUNCTIONS

O 2
max — min
a Amplitude =
2

b Period = 12 Take the horizontal distance between
c4
maximum points as your period: 16 — 4 = 12
max + min 2
d Vertical shift = The function y = cos x would have a maximum
where x = 0. This function has a maximum at
2
x = 4. Hence a horizontal shift of 4 to the right.
e m = 3 cos +5
You can find the vertical shift by looking at the
max + min

graph or by using the formula

2
Put all of the answers together using

m = a cos (b(h — c)) + d

+5 = 2.10m Substitute x = 9.5 hours into your equation.

8 onaooonaoo Graph your function and m = 4 on the same

6 axes using technology. Show a sketch as
4
2 working to get m = 12.35 hours. Multiply

x 0.35 by 60 to convert the decimal number

2 4 6 8 10 12 14 16 18 20 22 24 of hours into minutes.

12:21 pm

Exercise 12J

1 The height, h metres, of the tide on 2 The number of visitors (y) to a ski

New Year's Day in Perth, Australia resort in Switzerland is modelled by the

can be modelled using the function function y = 3000 — l)) + 10 000,

Ath(t) =5sin where x is the number of months after I

—5) +7, where t is the January.

number of hours after midnight. a What is the maximum number of
visitors and when does this occur?
a Find the maximum and minimum
b What is the minimum number of visitors
values for the depth of water. and when does this occur?

b What time is high tide? C How many visitors are at the resort on I
C What time is low tide?
d What is the depth of the water at 9.00 am? May?

e Find all the times during a 24-hour

period when the depth of the water is

532

12.4

3 This graph shows the number of fish in a C Find the amplitude of the function.

lagoon over a 24-hour period starting at d Hence or otherwise, find the value of a
midnight. and of c.

20 e Write down the period of the function.
18
16 f Hence, find the value of b.
14
12 g Use technology to determine the first
value of t for which the end of the
8 blade is 30 metres above the ground.
6
4 S A Ferris wheel moves with constant speed
2
and completes one rotation every 40 seconds.
2 4 6 8 10 12 14 16 18 20 22 24
mThe wheel has a radius of 12 and its lowest
Hour mpoint is 2 above the ground.

a What was the maximum number of fish The height of a chair on the Ferris wheel
above ground can be modelled by the
in the lagoon? function, h(t) = a cos bt+ c, where t is the

b How many fish were in the lagoon at time in seconds.

midnight? a Find the value of a, b and c.
c Find an equation for this function.
mThe chair first reaches a height of 20
d How many fish were in the lagoon at
above the ground after p seconds.
2pm?
4 The height, h(t), in metres, of the end of a b Find the angle that the chair has rotated
through to reach this position.
windmill blade above the ground at time,
t seconds, is a cosine function of the form c Find the value of p.
h(t) = a cos bt + c. The following graph
shows one rotation of the blade. m6 A Ferris wheel with a radius of 20 rotates

35 once every 40 seconds. Passengers get on

30 mI above level ground.

25 a Find a cosine equation for the height of
a seat on the Ferris wheel.
g 20
b Find the length of time during one
15 rotation that a passenger on the Ferris

10 mwheel will be at least 23 above the

oxvza ground.

5 In a physics experiment, a weight is
attached to the end of a long spring and
12
Time (s) bounces up and down. As it bounces, its
distance from the floor can be modelled
a Write down the maximum height of the with a cosine function. The weight is

end of blade above the ground. released from a high point 60 cm above the

b Write down the minimum height of the floor at 0.3 seconds and first reaches its low

end of blade above the ground. point, 40 cm above the floor at 1.8 seconds.

a Write an equation to model the distance
of the weight from the floor in seconds.

b Find the distance of the weight from the

floor at 17.2 seconds.

C Find the time when the weight was first
59 cm above the floor.

533

12 PERIODIC RELATIONSHIPS: TRIGONOMETRIC FUNCTIONS

Developing your toolkit

Now do the Modelling and investigation activity on page 540.

Chapter summary 1 radian

• A radian is the measure of the angle with its vertex at the centre of the circle O
between two radii with endpoints that are one radius length apart on the

circumference of the circle.

• 2n radians = 3600 or 1 radian = 5?.30.

• Converting between degrees and radians:

Degrees to radians: Radians to degrees:
Multiply the degree measure by: Multiply the radian measure by:

1800 1800
• The length of an arc is given by:

and the area of a sector by: arc
sector
A = —r20

2

where Ois measured in radians.

segment

• The unit circle is a circle with its centre
at the origin and of radius 1 unit.

(-1,0 1

(1,0)

x

(0,-1)

Trigonometric ratios

sin sin( 180 — 9)
cosØ= cos(—0)
cose= cos(360 — 9)
tan tan(180 + 0)

534

12

Pythagorean identity HINT

sin29+ cos2Ø= I You should keep the formula booklet

Tangent identity on your table when you are working to
see what is in there to help you, like
tano = sino trigonometric identities, and what is not

cos0 there that you will have to remember, like
The double angle identities the exact values of trigonometric ratios

sin 2 sin ecos 6432of 0, —,—, —, and their multiples,

cos20= I — 2 sin2 0 including the relationship between

cos29= 2 cos2 0— I angles in different quadrants.
cos20= cos2 0— sin2Ø
The tangent curve
The sine and cosine curves

1 x
0.5

-0.5
-1

• y=—sinx or y= —cosxisa reflection in the x-axis.

y=sin(—x) or is a reflection in they-axis.

• y=sin(x— c) or translates the function c units horizontally.

x+y = sin d or y = cos x + d translates the function d units vertically.

• y=asin b(x— c)+d

max — min
lal is the amplitude. Amplitude =

27t 2

period =
b

c is the horizontal shift and

d is the vertical shift. Vertical shift = max + min

2

Developing inquiry skills

Can you find the equation of the function from the opening problem?
Would you use sine or cosine?
What would be the difference if you used sine and your friend used
cosine?

535

12 PERIODIC RELATIONSHIPS: TRIGONOMETRIC FUNCTIONS

Chapter review Click here for a mixed
review exercise

1 Rewrite each angle in radians as a 8 a Given that 2sin2Ø+ sine— I = 0, find
the two values for sin 9.
multiple of n.

a 300 b 1500 c 3150 b Given that 0 2m, find three

possible values for e.

d 1200 e -200 -2400 3

g ¯2700 1440 9 Given that sino = where 9is an obtuse
4

angle, find:

2 Rewrite each angle in degrees.

a cose b cos 2 9

37t 77t 77t d 10 This graph is the function y = acos bx.
9
a b c 30
2 6 12 h
15 20 nouuou
e 111t 111t
3 10
30 6 x

3 a Find the measure of the indicated -10 ooaooa
-20 omoouo
central angle in radians
-30
b Find the area of the minor sector.
Find the values of a and b.
6

5

Given that sin 9=0.6, and 0 is acute, 11 This graph shows the function
find cos and tan 9. f(x) = asinbx, where b > 0. Find

b Solve 2sinx = tan x for 2 x
0
2

S Solve for 0 2m giving your answer in

radians as multiples of rt.

a sino = — 1 b cos0 =

22

c tan I

6 The expression 8 sin x cosx can be a Find the period of f.
expressed in the form a sin bx.
b Write down the amplitude of f.
a Find the values of a and of b. C Write down the value of a.

b Hence or otherwise, solve the equation d Find the value of b.

6 sinxcosx= 2, 0 27t

—12 f(x) = 5 sin for 0 4.

? Given that —SO 12 2

2 and cos0 — find a Write down the amplitude of f.

13

each value. b Find the period of f.

a sine b COS 29 c sin (9+ z) c Sketch the function.

536

12

13 OABC is a sector of a circle with radius 8 cm. O18 This circle has centre and radius 8 cm.

Find the shaded area when 0 = The points A, B and C lie on the circle. O,

6 C, and D lie on a straight line. Angle
ADC = 0.4 radians and angle AOC = 0.8

radians. Find:

c

8 cm 8 cm o.

Dc

14 This diagram shows a triangle ABC and a a AD
sector BDC of a circle with centre B and b OD
radius 4 cm. The points A B and D are on c the area of sector OABC
the same line and AB = Gcm, BC = 4 cm,
the area of triangle ABC is 2 cm2 and angle d the area of region ABCD.
ABC is obtuse. Find:
19 The function fis defined by
c
—f(x) = 20 sin 3xcos 3x, 0 x
4cm
3
a the angle ABC
a Write down an expression for f(x) in the
b the exact area of the sector BDC.
form a sin bx.

b Solve f(x) = 0, giving your answers in

terms of rt.

20 Given f(0) = 2 cos 20 + cosØ+ 3, for

15 Use a sketch to help you solve -3600 3600,

a cosx = x2 for 0 x— a show that this function may be written
as f(Ø) = cos2Ø + cos 0+ l.
b 2
b how many distinct values of cos 9 satisfy
for 0.5 S x < 1.5.
this equation?
16 In triangle ABC, AB is 10 cm, BC is 8 cm
and angle ABC is obtuse. The area of the c find all values of 0 which satisfy this

triangle is 10 cm2 equation.

Find the size of angle ABC in radians. 21 A Ferris wheel with centre O and a radius

1? This circle has a radius 10 cm with centre O. mof 15 is represented in the diagram.

The points A and B are on the circle, and AWhen seat is at ground level, another
angle AOB is 0.8 radians. Angle ONA is
seat is at B, where angle AOB = It
a right angle. Find the area of the shaded

region.

IOcm 15m

0.8 6

c

a Find the length of the arc AB.

12 PERIODIC RELATIONSHIPS: TRIGONOMETRIC FUNCTIONS

b Find the area of the sector AOB a Find the value of a.
c Find the height of A above the ground
b Show that b = E.
when the wheel turns clockwise
4
through an angle of 27t
c Find the value of d.
3
3
The height, h metres, of seat C above the
ground after t minutes, can be modelled by The graph of fis translated by and then

the function reflected in the x-axis to give the graph of g.
d Find the equation of g(x).
h(t) = 15-15cos2 t+—
Exam-style questions
8
24 Pi: Let A (t) = 2cos2 t— 3cost + l, 0 S t 2m.
—d Find the height of seat C when t =
a Factorize A(t). (2 marks)
4
e Find the initial height of seat C b Hence, solve the equation A(t) = 0
f Find the time at which seat C first
for 0 t 2;t. (4 marks)
reaches its highest point.
25 Pl: Solve the equation 2cos2 x = sin 2x for
22 This graph shows the functions
f(x) = pcos(qx) and g(x) = sin(bx) — d 0 t 2m, giving your answer in terms

4 of n. (5 marks)
3
26 Pl: A trigonometric function defined by
2
f (x) = asin (bx) + c, where a, b, c e R.
1
The period of the function fis n, the maxi-
x
mum offis 14 and the minimum is 8.
-1
a Find the values of the parameters
-2 (4 marks)

-3 b Hence, sketch the graph of f.
-4 (3 marks)

a Find the values of p, q, b and d. 27 Pi: Let S(x) = (sin 2x + cos2x)2

b Solve the equationf(x) =g(x) for a Show that S (x) = sin (4x) +1, for all

23 This graph shows the function (3 marks)
f(x) = asin(b x) + d for 0 S x 12. There is a
b Hence, sketch the graph of S for
maximum point at (2, 8) and a minimum
0 SXS1t. (3 marks)
point at (6, 2).
C State
f(x)
i the period of S
10
ii the range of S. (2 marks)
8
6 d Sketch the graph of the function C
4
2 defined by C (x) = cos(2x) — l, for

x (3 marks)

2 4 6 8 10 12 The graph of C can be obtained from

538 the graph of S under a horizontal

Kstretch of scale factor followed by a

translation by the vector

e Write down q

i the value of k

ii a possible vector (3 marks)
q