CHAPTER 10 Pythagoras’ theorem and trigonometry (2) Linear … · 2010-07-18 · CHAPTER 10 Linear equations In Chapter 19, Pythagoras’theorem and trigonometry were used to find

498

CHAPTER 10 Linear equations

In Chapter 19, Pythagoras’ theorem and trigonometry were used to find the lengths of sides and the sizesof angles in right-angled triangles.These methods will now be used with three-dimensional shapes.

31.1 Problems in three dimensionsIn a cuboid all the edges are perpendicular to each other.Problems with cuboids and other 3-D shapes involve identifying suitable right-angled triangles and applying Pythagoras’ theorem and trigonometry to them.

ABCDEFGH is a cuboid with length 8 cm, breadth 6 cm and height 9 cm.

a i Calculate the length of AC.

ii Calculate the length of AG.Give your answer correct to 3 significant figures.

b Calculate the size of angle GAC.Give your answer correct to the nearest degree.

Solution 1a i

AC2 � AB2 � BC2

� 82 � 62

� 64 � 36

� 100

AC � 10 cm

ii

AG2 � AC2 � CG2

AG2 � 102 � 92

� 100 � 81

� 181

AG � �181� � 13.4536 …

AG � 13.5 cm (to 3 s.f.)

10 cm

9 cm

A C

G

8 cm

6 cm

A B

C

Example 1

31C H A P T E R

Pythagoras’ theoremand trigonometry (2)

8 cm

6 cm

9 cm

E

H G

F

C

BA

D

Look for a right-angled triangle where AC is one side and the lengths of theother two sides are known.

ABC is a suitable triangle.

So draw triangle ABC marking the known lengths.

Look for a right-angled triangle where AG is one side and the lengths of theother two sides are known.

ACG is a suitable triangle.

So draw triangle ACGmarking the known lengths.

Use Pythagoras’ theorem for this triangle.

Use Pythagoras’ theorem for this triangle.

A

C

G

499

31.1 Problems in three dimensions CHAPTER 31

b

Exercise 31A

Where necessary give lengths correct to 3 significant figures and angles correct to one decimal place.

1 ABCDEFGH is a cuboid of length 8 cm, breadth 4 cm and height 13 cm.

a Calculate the length ofiii AC ii GBiii FA iv GA.

b Calculate the size ofi angle FAB ii angle GBC iii angle GAC.

2 ABCDEF is a triangular prism.In triangle ABC angle CAB � 90°,AB � 5 cm and AC � 12 cm.In rectangle ABED the length of BE � 15 cm.

a Calculate the length of CB.

b Calculate the length ofi CE ii AF.

c Calculate the size ofi angle FED ii angle FAD.

3 The diagram shows a square-based pyramid.The lengths of sides of the square base, ABCD, are 10 cm and the base is on a horizontal plane.The centre of the base is the point M and the vertex of the pyramid is O, so that OM is vertical.The point E is the midpoint of the side AB.OA � OB � OC � OD � 15 cm.a Calculate the length of i AC ii AM.b Calculate the length of OM.c Calculate the size of angle OAM.d Hence find the size of angle AOC.e Calculate the length of OE.f Calculate the size of angle OAB.

Angle between a line and a planeImagine a light shining directly above AB onto the plane.AN is the shadow of AB on the plane.A line drawn from point B perpendicular to the plane will meet the line AN and form a right angle with this line.Angle BAN is the angle between the line AB and the plane.

10 cm

9 cm

A C

G For angle GAC.

9 cm is the opposite side.

10 cm is the adjacent side.

tan � �o

a

p

d

p

j�

�

C

BA

D

E F

H G

8 cm4 cm

13 cm

A B

C

F

D E

5 cm

12 cm

15 cm

O

M

A

BE

C

D

15 cm

10 cm

N

A

B

tan (angle GAC)� �190� � 0.9

angle GAC � 41.987 …°

Angle GAC � 42°

(to the nearest degree)

500

CHAPTER 31 Pythagoras’ theorem and trigonometry (2)

The diagram shows a pyramid.The base, ABCD, is a horizontal rectangle in which AB � 12 cm and AD � 9 cm.The vertex, O, is vertically above the midpoint of the base and OB � 18 cm.Calculate the size of the angle that OB makes with the horizontal plane.Give your answer correct to one decimal place.

Solution 2

DB2 � 92 � 122 � 81 � 144

DB2 � 225

DB � �225� � 15

MB � �12� DB � 7.5

cos (angle OBM) � �7

1

.

8

5�

angle OBM � 65.37 …°

The angle between OB and the horizontal plane is 65.4° (to one d.p.)

O

M B

18 cm

7.5 cm

D

A B12 cm

9 cm

O

M B

18 cm

O

A

M B

C

D

18 cm

9 cm12 cm

Example 2 O

A

B

C

D

18 cm

9 cm12 cm

The base, ABCD, of the pyramid is horizontal so the angle that OB makeswith the horizontal plane is the angle that OB makes with the base ABCD.

Let M be the midpoint of the base which is directly below O.

Join O to M and M to B.

As OM is perpendicular to the base of the pyramid the angle OBM is theangle between OB and the base and so is the required angle.

Draw triangle OBM marking OB � 18 cm.

To find the size of angle OBM find thelength of either MB or OM.

Calculate the length of MB which is �12� DB.

Draw the right-angledtriangle ABD marking theknown lengths.

Use Pythagoras’ theorem to calculate the length of DB.

For angle OBM, 18 cm is the hypotenuse, 7.5 cm is the adjacent side.

cos � �h

a

y

d

p

j�

501

31.1 Problems in three dimensions CHAPTER 31

Exercise 31B

Where necessary give lengths correct to 3 significant figures and angles correct to one decimal place.

1 The diagram shows a pyramid.The base, ABCD, is a horizontal rectangle in which AB � 15 cm and AD � 8 cm.The vertex, O, is vertically above the centre of the base and OA � 24 cm.Calculate the size of the angle that OA makes with the horizontal plane.

2 ABCDEFGH is a cuboid with a rectangular base in which AB � 12 cm and BC � 5 cm.The height, AE, of the cuboid is 15 cm.Calculate the size of the angle

a between FA and ABCDb between GA and ABCDc between BE and ADHEd Write down the size of the angle between HE and ABFE.

3 ABCDEF is a triangular prism.In triangle ABC, angle CAB � 90°, AB � 8 cm and AC � 10 cm.In rectangle ABED, the length of BE � 5 cm.Calculate the size of the angle between

a CB and ABEDb CD and ABEDc CE and ABEDd BC and ADFC.

4 The diagram shows a square-based pyramid.The lengths of sides of the square base, ABCD,are 8 cm and the base is on a horizontal plane.The centre of the base is the point M and the vertex of the pyramid is O so that OM is vertical.The point E is the midpoint of the side AB.OA � OB � OC � OD � 20 cmCalculate the size of the angle between OEand the base ABCD.

� O

A

B

C

D

15 cm

24 cm

8 cm

12 cm

5 cm

15 cm

E

H G

F

C

BA

D

5 cm

8 cm

10 cm

A B

E

F

C

D

O

M

A

BE

C

D

20 cm

8 cm

5 ABCD is a horizontal rectangular lawn in a garden and TC is a vertical pole. Ropes run from the top of the pole,T, to the corners, A, B and D, of the lawn.

a Calculate the length of the rope TA.

b Calculate the size of the angle made with the lawn by

i the rope TB ii the rope TD iii the rope TA.

6 The diagram shows a learner’s ski slope, ABCD, of length, AB, 500 m. Triangles BAF and CDEare congruent right-angled triangles and ABCD, AFED and BCEF are rectangles.The rectangle BCEF is horizontal and the rectangle AFED is vertical.The angle between AB and BCEF is 20° and the angle between AC and BCEF is 10°.

Calculate

a the length of FB b the height of A above Fc the distance AC d the width, BC, of the ski slope.

7 Diagram 1 shows a square-based pyramid OABCD.Each side of the square is of length 60 cm and OA � OB � OC � OD � 50 cm.

Diagram 2 shows a cube, ABCDEFGH, in which each edge is of length 60 cm.

A solid is made by placing the pyramid on top of the cubeso that the base, ABCD, of the pyramid is on the top,ABCD, of the cube.The solid is placed on a horizontal table with the face,EFGH, on the table.

a Calculate the height of the vertex O above the table.

b Calculate the size of the angle between OE and thehorizontal.

A

D

C

BF

E

500 m

502


A B

C

T

D

8 m

12 m

6 m

O

A

B

C

D

50 cm

60 cm60 cm

D

C

B

F

GA

E

H

60 cm60 cm

60 cm

Diagram 1

Diagram 2

31.2 Trigonometric ratios for any angleThe diagram shows a circle, centre the origin O and radius 1 unit. Imagine a line, OP, of length 1 unitfixed at O, rotating in an anticlockwise direction about O, starting from the x-axis.

The diagram shows OP when it has rotated through 40°.

The right-angled triangle OPQ has hypotenuse OP � 1

Relative to angle POQ, side PQ is the opposite side and side OQ is the adjacent side.This means that

OQ � cos 40° and PQ � sin 40°

For P, x � cos 40° and y � sin 40° so the coordinates of P are (cos 40°, sin 40°).

In general when OP rotates through any angle �°, the position of P on the circle, radius � 1 is givenby x � cos �°, y � sin �°.

The coordinates of P are (cos �°, sin �°).

So when OP rotates through 400° the coordinates of P are (cos 400°, sin 400°).A rotation of 400° is 1 complete revolution of 360° plus a further rotation of 40°.

The position of P is the same as in the previous diagram so (cos 400°, sin 400°) is the same point as(cos 40°, sin 40°), therefore cos 400° � cos 40° and sin 400° � sin 40°.

If OP rotates through �40° this means OP rotates through 40° in a clockwise direction.

�1

�1.2

�0.8

�0.6

�0.4

�0.2

0.2

O

0.4

0.6

0.8

1

y

�1�1.2 �0.8 �0.6 �0.4 �0.2 0.2 0.4 0.6 0.8 1

1

x

P

40° Q

503

31.2 Trigonometric ratios for any angle CHAPTER 31

504


For � � 136, � � 225, � � 304 and � � �40 the position of P is shown on the diagram.

For P when � � 136, x � cos 136° and y � sin 136°.From the diagram, cos 136° � 0 and sin 136° � 0



For P when � � �40, x � cos �40° and y � sin �40°.From the diagram, cos �40° � 0 and sin �40° � 0

The diagram shows for each quadrant whether the sine andcosine of angles in that quadrant are positive or negative.

The sine and cosine of any angle can be found using your calculator. The following table shows someof these values corrected where necessary to 3 decimal places.

Using these values and others from a calculator the graphs of y � sin �° and y � cos �° can bedrawn. A graphical calculator would be useful here.

�1

�1.2

�0.8

�0.6

�0.4

�0.2

0.2

O

0.4

0.6

0.8

1.2

1

y

�1�1.2 �0.8 �0.6 �0.4 �0.2 0.2 0.4 0.6 0.8 1.21 x

304°

P

P

225°

P

P

136°

�40°

� 0 30 40 45 60 90 136 180 225 270 304 360

sin �° 0 0.5 0.643 0.707 0.866 1 0.695 0 �0.707 �1 �0.829 0

cos �° 1 0.866 0.766 0.707 0.5 0 �0.719 �1 �0.707 0 0.559 1

sin �cos �

1st

4th

2nd

3rd

sin �cos �

sin �cos �

sin �cos �

505

31.2 Trigonometric ratios for any angle CHAPTER 31

Graph of y � sin �°

Graph of y � cos �°

Notice also that the graph of y � sin �° and the graph of y � cos �° are horizontal translations ofeach other.

To find the value of the tangent of any angle, use tan �° � �c

s

o

in

s

�

�

°

°�

From the graph of y � cos �° it can be seen that cos �° � 0 at � � 90, 270, 450, … for example.As it is not possible to divide by 0 there are no values of tan �° at � � 90, 270, 450, … that is, thegraph is discontinuous at these values of �.

Graph of y � tan �°

Notice also that tan �° can take any value.

�8

�4�6

�2O

4

8

2

6

y

�180 180 360 540 θ

�1

�0.5

O

0.5

1y

�180 180 360 540 θ

�1

�0.5

O

0.5

1y

�180 180 360 540 θ

Notice that the graph:● cuts the �-axis at … , �180, 0, 180, 360, 540, … ● repeats itself every 360°, that is, it has a period

of 360°● has a maximum value of 1 at � � … , 90, 450, …● has a minimum value of �1 at

� � … , �90, 270, …

Notice that the graph:● cuts the �-axis at … �90, 90, 270, 450, …● repeats itself every 360°, that is it has a period

of 360°● has a maximum value of 1 at � � … , 0, 360, …● has a minimum value of �1 at

� � … , �180, 180, 540, …

Notice that the graph:

● cuts the �-axis where tan �° � 0, that is, at … �180, 0, 180, 360, 540 …

● repeats itself every 180°, that is it has a periodof 180°

● does not have values at � � �90, �270, �450, …

● does not have any maximum or minimum points.

For values of � in the interval – 180 to 360 solve the equation

ii sin �° � 0.7

ii 5 cos �° � 2

Give each answer correct to one decimal place.

Solution 3i sin �° � 0.7

� � 44.4

� � 44.4, 180 � 44.4

� � 44.4, 135.6

ii 5 cos �° � 2

cos �° � �25� � 0.4

� � 66.4

� � 66.4, �66.4, 360 � �66.4

� � 66.4, �66.4, 293.6

Exercise 31C

1 For �360 � 360 sketch the graph of

a y � sin �° b y � cos �° c y � tan �°.

2 Find all values of � in the interval 0 to 360 for which

a sin �° � 0.5 b cos �° � 0.1 c tan �° � 1

3 a Show that one solution of the equation 3 sin �° � 1 is 19.5, correct to 1 decimal place.

b Hence solve the equation 3 sin �° � 1 for values of � in the interval 0 to 720

�1

O

1y

y � 0.4

�180 360180 θ

�1

O

1y

y � 0.7

�180 360180 θ

Example 3

506


Use a calculator to find one value of � .

To find the other solutions draw a sketchof y � sin �° for � from �180 to 360

The sketch shows that there are twovalues of � in the interval �180 to 360 for which sin �° � 0.7

One solution is � � 44.4 and by symmetrythe other solution is � � 180 � 44.4

Divide each side of the equation by 5

Use a calculator to find one value of � .

To find the other solutions draw a sketch ofy � cos �° for � from �180 to 360

The sketch shows that there are three values of � inthe interval �180 to 360 for which cos �° � 0.4

One solution is � � 66.4 and by symmetryanother solution is � � �66.4

Using the period of the graph the other solutionis � � 360 � �66.4

4 a Show that one solution of the equation 10 cos �° � �3 is 107.5 correct to 1 decimal place.

b Hence find all values of � in the interval �360 to 360 for which 10 cos �° � �3

5 Solve 4 tan �° � 3 for values of � in the interval �180 to 360

31.3 Area of a triangleLabelling sides and angles The vertices of a triangle are labelled with capital letters.The triangle shown is triangle ABC.

The sides opposite the angles are labelled so that a is the length of the side opposite angle A, b is thelength of the side opposite angle B and c is the length of the side opposite angle C.

Area of a triangle � �12� base height

Area of triangle ABC � �12� bh

In the right-angled triangle BCN h � a sin C

So area of triangle ABC � �12� b a sin C that is

The angle C is the angle between the sides of length a and b and is called the included angle.The formula for the area of a triangle means that

Area of a triangle � �12� product of two sides sine of the included angle.

For triangle ABC there are other formulae for the area.

Area of triangle ABC � �12� ab sin C � �

12� bc sin A � �

12� ac sin B.

These formulae give the area of a triangle whether the included angle is acute or obtuse.

Find the area of each of the triangles correct to 3 significant figures.

a b

118°

7.4 m

16.2 m

C A

B

37°

5.8 cm

7.3 cm

Example 4

area of triangle ABC � �12� ab sin C

507

31.3 Area of a triangle CHAPTER 31

C A

B

a c

b

C AN

B

ac

b

h

508


Solution 4

a Area � �12� 7.3 5.8 sin 37°

Area � 12.74 …

Area � 12.7 cm2

b Area � �12� 7.4 16.2 sin 118°

Area � 52.92 …

Area � 52.9 m2

The area of this triangle is 20 cm2.Find the size of the acute angle x°.Give your angle correct to one decimal place.

Solution 5�12� 8.1 6.4 sin x° � 20

sin x° � �8

2

.1

2

6

0

.4� � 0.7716

x° � 50.49 …°

x° � 50.5°

Exercise 31D

Give lengths and areas correct to 3 significant figures and angles correct to one decimal place.

1 Work out the area of each of these triangles.

i ii iii

iv v vi

2 ABCD is a quadrilateral.Work out the area of the quadrilateral.

13.4 cm8.6 cm 148.6°

4.7 cm

9.6 cm

137°

4.6 cm 4.6 cm76.3°

10.6 cm

9.1 cm

34.7°

13.5 cm9.2 cm

28°

6.9 cm

9.3 cm

43°

Example 5

Substitute a � 7.3 cm, b � 5.8 cm, C � 37° into area � �12� ab sin C

Give the area correct to 3 significant figures and state the units.

Substitute into area of a triangle � �

12� product of two sides sine of the included angle.

Use area of a triangle � �

12� product of two sides sine of the included angle.

Find the value of sin x°.

Give the angle correct to one decimal place.

6.4 cm8.1 cm

x°

8.6 cm

12.6 cm9.4 cm

57°

80°

A B

C

D

3 The area of triangle ABC is 15 cm2

Angle A is acute.Work out the size of angle A.

4 The area of triangle ABC is 60.7 m2

Work out the length of BC.

5 a Triangle ABC is such that a � 6 cm, b � 9 cm and angle C � 25°.Work out the area of triangle ABC.

b Triangle PQR is such that p � 6 cm, q � 9 cm and angle R � 155°.Work out the area of triangle PQR.

c What do you notice about your answers? Why do you think this is true?

6 The diagram shows a regular octagon with centre O.

a Work out the size of angle AOB.OA � OB � 6 cm.

b Work out the area of triangle AOB.

c Hence work out the area of the octagon.

7 Work out the area of the parallelogram.

8 a An equilateral triangle has sides of length 12 cm.Calculate the area of the equilateral triangle.

b A regular hexagon has sides of length 12 cm.Calculate the area of the regular hexagon.

9 The diagram shows a sector, AOB, of a circle, centre O.The radius of the circle is 8 cm and the size of angle AOB is 50°.

a Work out the area of triangle AOB.

b Work out the area of the sector AOB.

c Hence work out the area of the segment shown shaded inthe diagram.

63°12.8 cm

5.7 cm

509

31.3 Area of a triangle CHAPTER 31

8.4 cm

6.5 cm

A B

C

12.6 m

35°A B

C

A B

O

50°8 cm

8 cm

A

B

O

510


31.4 The sine rule

The last section showed that

�12� ab sin C � �

12� bc sin A and �

12� bc sin A � �

12� ca sin B

cancelling �12� and b from both sides cancelling �

12� and c from both sides

a sin C � c sin A and b sin A � a sin B

or or

�sin

aA

� � �sin

cC

� and �sin

bB

� � �sin

aA

�

Combining these results

This result is known as the sine rule and can be used in any triangle.

Using the sine rule to calculate a length

Find the length of the side marked a in the triangle.Give your answer correct to 3 significant figures.

Solution 6

�sin

a

37°� � �

sin

8.

7

4

4°�

a ��8.4

s

in

s

7

in

4°

37°�

a � 5.258 …

a � 5.26 cm

Example 6

�sin

aA

� � �sin

bB

� � �sin

cC

�

Area of triangle ABC � ab sin C � bc sin A � ca sin B12

12

12

b

ca

C A

B

74°

37°

a

8.4 cm

Substitute A � 37°, b � 8.4, B � 74° into �sin

a

A� � �

sin

b

B�.

Multiply both sides by sin 37°.

511

31.4 The sine rule CHAPTER 31

Find the length of the side marked x in the triangle.Give your answer correct to 3 significant figures.

Solution 7Missing angle � 180 � (18 � 124)

� 38°

�sin 1

x

24°� � �

sin

9.

3

7

8°�

x ��9.7

sin

si

3

n

8

1

°

24°�

x � 13.06 …

x � 13.1 cm

Using the sine rule to calculate an angleWhen the sine rule is used to calculate an angle it is a good idea to turn each fraction upside down(the reciprocal). This gives

�sin

a

A� � �

sin

b

B� � �

sin

c

C�

Find the size of the acute angle x in the triangle.Give your answer correct to one decimal place.

Solution 8

�s

7

in

.9

x� � �

sin

8.

7

4

4°�

sin x ��7.9

8

s

.4

in 74°�

sin x � 0.904 …

x � 64.69 …°

x � 64.7°

Example 8

124°38°

18°

x9.7 cm

Example 7

The angle opposite 9.7 cm must be found before the sine rule can be used.

Use the angle sum of a triangle.

Write down the sine rule with x opposite 124° and 9.7 opposite 38°.

Multiply both sides by sin 124°.

Write down the sine rule with x opposite 7.9 and 74° opposite 8.4

Multiply both sides by 7.9

Find the value of sin x.

124°

18°

x9.7 cm

74°

x

8.4 cm

7.9 cm

Exercise 31E

Give lengths and areas correct to 3 significant figures and angles correct to 1 decimal place.

1 Find the lengths of the sides marked with letters in these triangles.

a b c

d e f

2 Calculate the size of each of the acute angles marked with a letter.

a b c d

3 The diagram shows quadrilateral ABCDand its diagonal AC.

a In triangle ABC, work out the length of AC.b In triangle ACD, work out the size of angle DAC.

c Work out the size of angle BCD.

4 In triangle ABC, BC � 8.6 cm, angle BAC � 52° and angle ABC � 63°.

a Calculate the length of AC.

b Calculate the length of AB.

c Calculate the area of triangle ABC.

5 In triangle PQR all the angles are acute. PR � 7.8 cm and PQ � 8.4 cm.Angle PQR � 58°.

a Work out the size of angle PRQ.

b Work out the length of QR.

6 The diagram shows the position of a port (P),a lighthouse (L) and a buoy (B).The lighthouse is due east of the buoy.The lighthouse is on a bearing of 035° from the portand the buoy is on a bearing of 312° from the port.

a Work out the size of i angle PBL ii angle PLB.

The lighthouse is 8 km from the port.

b Work out the distance PB.

c Work out the distance BL.

d Work out the shortest distance from the port (P) to the line BL.

104°

C

7.6 cm

18.4 cm

21° B

17 cm 6.8 cm

32°A

8 cm6 cm

22°

f g113°

14.9 cm

17°

134°

e

6.1 cm

76°

64°d 13.6 cm

62°

58°c

6.1 cm

51°27°

b 4.2 cm79°

46°

a

11 cm

512


73°

E D12.7 cm

9.1 cm

102°

46°18°

A

DC

B

8.6 cm

5.7 cm

P

B L

N

513

31.5 The cosine rule CHAPTER 31

31.5 The cosine ruleThe diagram shows triangle ABC.The line BN is perpendicular to AC and meets the line AC at N so that AN � x and NC � (b � x).The length of BN is h.

In the right-angled triangle ANB, x � c cos A

Using 2

This result is known as the cosine rule and can be used in any triangle.

Using the cosine rule to calculate a length

Find the length of the side marked with a letter in each triangle.Give your answers correct to 3 significant figures.

a b

Solution 9a a2 � 122 � 82 � 2 12 8 cos 24°

a2 � 144 � 64 � 175.4007 …

a2 � 32.599 27 …

a � �32.599� 27 …�a � 5.709 577 …

a � 5.71 cm

b x2 � 7.32 � 5.82 � 2 7.3 5.8 cos 117°

x2 � 53.29 � 33.64 � 84.68 (�0.4539 …)

x2 � 86.93 � 38.44 …

x2 � 125.37 …

x � �125.37� …�x � 11.19 …

x � 11.2 cm

117°

5.8 cm

7.3 cmx

8 cm

24°12 cm

B

C

a

A

Example 9

b2 � a2 � c2 � 2ac cos Bc2 � a2 � b2 � 22ab cos C

Similarly

and

a2 � b2 � c2 � 2bc cos A

AN

b

ahc

C

BIn triangle ANBPythagoras’ theoremgivesc2 � x2 � h2…

In triangle BNCPythagoras’ theoremgivesa2 � (b � x)2 � h2

a2 � b2 � 2bx � x2 � h2

Using substitute c2 for x2 � h2

a2 � b2 � 2bx � c2…

x (b � x)

11

2

Substitute b � 12, c � 8, A � 24° into a2 � b2 � c2 � 2bc cos A.

Evaluate each term separately.

Take the square root.

Substitute the two given lengths and the includedangle into the cosine rule.

Take the square root.

cos 117° � 0

Using the cosine rule to calculate an angleTo find an angle using the cosine rule, when the lengths of all three sides of a triangle are known,rearrange a2 � b2 � c2 � 2bc cos A.

2bc cos A � b2 � c2 � a2

cos A ��b2 �

2

cb

2

c� a2

�

Similarly cos B ��a2 �

2

ca

2

c� b2

�

and cos C ��a2 �

2

ba

2

b� c2

�

Find the size of a angle BAC b angle X. Give your answers correct to one decimal place.

a b

Solution 10

a cos A ��11

2

2

�

1

1

1

62

�

1

1

6

32

�

cos A � �2

3

0

5

8

2�

cos A � 0.590 909 …

A � 53.77 …°

A � 53.8°

b cos X ��8.6

2

2 �

6

8

.

.

9

6

2

�

6

1

.9

2.72

�

cos X � ��

11

3

8

9

.

.

6

7

8

2�

cos X � �0.334 68 …

X � 109.55 …°

X � 109.6°

12.7 cm8.6 cm

6.9 cm

X

11 cm

13 cm16 cm

B

A C

Example 10

514


Substitute b � 11, c � 16, a � 13 into cos A ��b2 �

2

c

b

2

c� a2

�.

Substitute the three lengths into the cosine rule noting that 12.7 cm isopposite the angle to be found.

The value of cos X is negative so X is an obtuse angle.

Exercise 31F

Where necessary give lengths and areas correct to 3 significant figures and angles correct to1 decimal place.

1 Calculate the length of the sides marked with letters in these triangles.

a b c

d e f

2 Calculate the size of each of the angles marked with a letter in these triangles.

a b

c d

3 The diagram shows the quadrilateral ABCD.

a Work out the length of DB.

b Work out the size of angle DAB.

c Work out the area of quadrilateral ABCD.

4 Work out the perimeter of triangle PQR.

5 In triangle ABC, AB � 10.1 cm, AC � 9.4 cm and BC � 8.7 cm.Calculate the size of angle BAC.

6 In triangle XYZ, XY � 20.3 cm, XZ � 14.5 cm and angle YXZ � 38°.Calculate the length of YZ.

P

Q

R

8.6 cm

10.9 cm27°

D

8.6 cm

7.2 cm14.4 cm

C

6.8 cm

8.7 cm8.7 cm

B13.6 cm

9.4 cm15.3 cm

A11 cm

9 cm7 cm

8.4 cm

8.4 cm

147°

f

10.2 cm

6.3 cm

134°

e9.6 cm

9.6 cm 52°

d

18°16.2 cm

15.5 cm

c

75°

11.3 cmb

9.2 cm62°

8 cm

9 cm

a

515

31.5 The cosine rule CHAPTER 31

D

AB

C26.4 cm

16.3 cm

8.4 cm

56°

9.8 cm

516


7 AB is a chord of a circle with centre O.The radius of the circle is 7 cm and the length of the chord is 11 cm.Calculate the size of angle AOB.

8 The region ABC is marked on a school field.The point B is 70 m from A on a bearing of 064°.The point C is 90 m from A on a bearing of 132°.

a Work out the size of angle BAC.

b Work out the length of BC.

9 Chris ran 4 km on a bearing of 036° from P to Q. He then ran in a straight line from Q to Rwhere R is 7 km due East of P. Chris then ran in a straight line from R to P.Calculate the total distance run by Chris.

10 The diagram shows a parallelogram.Work out the length of each diagonal of the parallelogram.

31.6 Solving problems using the sine rule, the cosine rule and�12� ab sin C

The area of triangle ABC is 12 cm2

AB � 3.8 cm and angle ABC � 70°.

a Find the length of i BC ii AC.Give your answers correct to 3 significant figures.

b Find the size of angle BAC.Give your answer correct to 1 decimal place.

Solution 11a i �

12� 3.8 BC sin 70° � 12

BC ��3.8

2

sin

1

7

2

0°�

BC � 6.721 …

BC � 6.72 cm

ii b2 � 6.721 …2 � 3.82 � 2 6.721… 3.8 cos 70°

b2 � 59.613 … � 17.470 …

b2 � 42.142 …

b � 6.491 …

AC � 6.49 cm

Example 11

O

A

B7 cm

11 cm

A

C

B

90 m

70 m

N

8 cm

6 cm

65°

A

B C

3.8 cm

70°

Substitute c � 3.8, B � 70° into area � �12� ac sin B.

Substitute a � 6.721 …, c � 3.8 and B � 70°into b2 � a2 � c2 � 2ac cos B.

517

31.6 Solving problems using the sine rule, the cosine rule and �12�ab sin C CHAPTER 31

b �6.

s

7

in

21

A

…� � �

6

s

.

i

4

n

9

7

1

0

…

°�

sin A ��6.721

6

…

.49

1

s

…

in 70°�

sin A � 0.9728 …

A � 76.62 …°

Angle BAC � 76.6°

Exercise 31G

Where necessary give lengths and areas correct to 3 significant figures and angles correct to1 decimal place, unless the question states otherwise.

1 A triangle has sides of lengths 9 cm, 10 cm and 11 cm.

a Calculate the size of each angle of the triangle.

b Calculate the area of the triangle.

2 In the diagram ABC is a straight line.

a Calculate the length of BD.

b Calculate the size of angle DAB.

c Calculate the length of AC.

3 The area of triangle ABC is 15 cm2.AB � 4.6 cm and angle BAC � 63˚.

a Work out the length of AC.

b Work out the length of BC.

c Work out the size of angle ABC.

4 ABCD is a kite with diagonal DB.

a Calculate the length of DB.

b Calculate the size of angle BDC.

c Calculate the value of x.

d Calculate the length of AC.

5 Kultar walked 9 km due South from point A to point B.He then changed direction and walked 5 km to point C.Kultar was then 6 km from his starting point A.

a Work out the bearing of point C from point B. Give your answer correct to the nearest degree.

b Work out the bearing of point C from point A. Give your answer correct to the nearest degree.

6 The diagram shows a pyramid. The base of the pyramid, ABCD, is a rectangle in which AB � 15 cm and AD � 8 cm.The vertex of the pyramid is O where OA � OB � OC � OD � 20 cm.Work out the size of angle DOB correct to the nearest degree.

Substitute a � 6.721 …, b � 6.491 … and B � 70° into �sin

a

A� � �

sin

b

B�.

D

A 5.4cm B C

12.9 cm 12 cm

65° 47°

D

B

A C

x cm

x cm8 cm

8 cm

50° 30°

D

A

B

C

O

8 cm15 cm

20 cm

7 The diagram shows a vertical pole, PQ, standing on a hill.The hill is at an angle of 8° to the horizontal.The point R is 20 m downhill from Q and the line PR is at 12° to the hill.

a Calculate the size of angle RPQ.

b Calculate the length, PQ, of the pole.

8 A, B and C are points on horizontal ground so that AB � 30 m, BC � 24 m and angle CAB � 50°.AP and BQ are vertical posts, where AP � BQ � 10 m.

a Work out the size of angle ACB.

b Work out the length of AC.

c Work out the size of angle PCQ.

d Work out the size of the angle between QC and the ground.

Chapter summary

Chapter 31 review questions1 In the diagram, XY represents a vertical tower

on level ground.A and B are points due West of Y.The distance AB is 30 metres.The angle of elevation of X from A is 30º.The angle of elevation of X from B is 50º.Calculate the height, in metres, of the tower XY.Give your answer correct to 2 decimal places.

(1384 June 1996)

2 The diagram shows triangle ABC.AC � 7.2 cm BC � 8.35 cmAngle ACB � 74°.

a Calculate the area of triangle ABC.Give your answer correct to 3 significant figures.

b Calculate the length of AB.Give your answer correct to 3 significant figures. (1385 June 2002)

518


RQ

P

8°

12°20 m

P Q

A B

C

50°

24 m

10 m10 m

30 m

You should now be able to:

use Pythagoras’ theorem to solve problems in 3 dimensions

use trigonometry to solve problems in 3 dimensions

work out the size of the angle between a line and a plane

draw sketches of the graphs of y � sin x°, y � cos x°, y � tan x° and use these graphs tosolve simple trigonometric equations

use the formula area � �12� ab sin C to calculate the area of any triangle

use the sine rule �sin

a

A� � �

sin

b

B� � �

sin

c

C� and the cosine rule a2 � b2 � c2 � 2bc cos A in

triangles and in solving problems.

�

�

�

�

�

�

X

YBA30°

30 m50°

Diagram NOTaccurately drawn

74°

7.2 cm 8.35 cm

C

BA


3 In triangle ABCAC � 8 cmCB � 15 cmAngle ACB � 70°.

a Calculate the area of triangle ABC.Give your answer correct to 3 significant figures.

X is the point on AB such that angle CXB � 90°.

b Calculate the length of CX.Give your answer correct to 3 significant figures. (1387 June 2003)

4 The diagram shows a cuboid.A, B, C, D and E are five vertices of the cuboid.AB � 5 cmBC � 8 cmCE � 3 cm.

Calculate the size of the angle the diagonal AE makes with the plane ABCD.Give your answer correct to 1 decimal place.

5 In triangle ABCAC � 8 cm BC � 15 cmAngle ACB � 70°.a Calculate the length of AB.

Give your answer correct to 3 significant figures.

b Calculate the size of angle BAC.Give your answer correct to 1 decimal place. (1387 June 2003)

6 This is a sketch of the graph of y � cos x°for values of x between 0 and 360.Write down the coordinates of the point

ii Aii B.

7 Angle ACB � 150°BC � 60 m.The area of triangle ABC is 450 m2

Calculate the perimeter of triangle ABC.Give your answer correct to 3 significant figures.

(1385 November 2000)

519

Chapter 31 review questions CHAPTER 31

8 cm

70°

AX

BC 15 cm


A B

3 cm

8 cm

5 cm

CD

E Diagram NOTaccurately drawn

70°

15 cm

8 cm

B C

A

360

A

B

y

O x

150°

A

60 m

B

C Diagram NOTaccurately drawn


8 The diagram shows a quadrilateral ABCD.AB � 4.1 cmBC � 7.6 cmAD � 5.4 cmAngle ABC � 117°Angle ADC � 62°.

a Calculate the length of AC.Give your answer correct to 3 significant figures.

b Calculate the area of triangle ABC.Give your answer correct to 3 significant figures.

c Calculate the area of the quadrilateral ABCD.Give your answer correct to 3 significant figures. (1385 June 2000)

9 This is a graph of the curve y � sin x° for 0 x 180

a Using the graph or otherwise, find estimates of the solutions in the interval 0 x 360 ofthe equationi sin x° � 0.2 ii sin x° � �0.6.

cos x° � sin (x � 90)° for all values of x.

b Write down two solutions of the equation cos x° � 0.2 (1385 November 2002)

10 In the diagram, ABCD, ABFE and EFCD are rectangles.The plane EFCD is horizontal and the plane ABFE is vertical.EA � 10 cmDC � 20 cmED � 20 cm.

Calculate the size of the angle that the line AC makes with the plane EFCD.

11 In triangle ABCAB � 10 cmAC � 14 cmBC � 16 cm.

a Calculate the size of the smallest angle in the triangle.Give your answer correct to the nearest 0.1°.

b Calculate the area of triangle ABC.Give your answer correct to 3 significant figures.

�1

�0.5

O

0.5

1

y

45 90 135 180 x

520


117° 62°

7.6 cm

4.1 cm 5.4 cm

B

A

D

C


E

AB

D

F

C

10 cm

20 cm20 cm

BC

A

10 cm14 cm

16 cm


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