HKDSE MATHEMATICS
Ronald Hui
Tak Sun Secondary School
HOMEWORK
SHW6-01, 6-A1
Deadline: 11 Jan 2016
No more delay please
22 October 2015
Ronald HUI
SUMMARY OF 2D TRIGONOMETRY
Ronald Hui
Tak Sun Secondary School
Similarly, for any △ABC, we have:
1
ab sin C
2
1
= bc sin A
2
1
= ac sin B
2
the included angle
of a and b.
Area of △ABC =
A
b
c
B
a
C
the included angle
the included angle
of b valid
and cfor
. right-angled triangles.
Note: The above formula is also
of a and c.
When C = 90,
area of △ABC 
1
1
1
ab sin C  ab sin 90  ab
2
2
2
Heron’s formula
Area of △ABC  s(s  a)(s  b)(s  c) ,
where s 
abc
.
2
s is called the semi-perimeter.
B
A
b
c
a
C
The sine formula
A
In △ABC,
a
b
c
=
=
sin A
sin B sin C
c
B
It can also be written as
sin B
sin C
sin A
a = b = c
b
a
Note: By the sine formula, we have:
1.
a
b
b
c
a
c

,

and

sin A sin B sin B sinC
sin A sinC
2.
sin A : sin B : sin C = a : b : c
C
In fact, for any △ABC, we have
c2 = a2 + b2  2ab cos C
Similarly, we can prove that
b2 = a2 + c2  2ac cos B
and a2 = b2 + c2  2bc cos A.
The above results are known as the cosine formula.
The cosine formula
In △ABC,
A
a2 = b2 + c2  2bc cos A,
b2 = a2 + c2  2ac cos B,
c2 = a2 + b2  2ab cos C.
b
c
B
a
C
The cosine formula can also be written as follows:
In △ABC,
b2  c 2  a2
cos A 
,
2bc
a2  c 2  b2
cos B 
,
2ac
a2  b2  c 2
cos C 
.
2ab
C
a
b
A
c
B
The cosine formula is also known as the cosine law or the
cosine rule.
In the figure, CB and AD are horizontal lines.
C
horizontal line
B
angle of depression of A from B
angle of elevation of B from A
A
horizontal line
D
We can see that:
Angle of elevation Angle of depression
=
of B from A
of A from B
alt. s, AD // CB
True Bearing and Compass Bearing
To
describe
the direction of a point relative to another
True
Bearing
point,
true bearing or compass bearing may be used.
N
 Directions are measured from the
P
north in a clockwise direction.

O
It is expressed as x, where
(i) 0  x < 360,
(ii) the integral part of x must
consist of 3 digits.
N
Refer to the figure on the right,
58
the true bearing of P from O is 058,
P
O
the true bearing of Q from O is 220.
40
Q
180 + 40
= 220
True Bearing and Compass Bearing
Compass Bearing
N60W
N
N40E
P
Q
60 40
W
S
E
O
70
25
R
S
S70W
S25E

Directions are
measured from the
north or the south.

It is expressed as
NxE, NxW,
SxE or SxW,
where 0 < x < 90.
Book 5A Chapter 6
Basic Terminologies in
3-dimensional Problems
Angle between Two Intersecting Straight Lines
The figure shows two non-parallel straight lines AB
and CD. They intersect each other at a point E.
D
A

C
E

The acute angle  formed is
called the angle between
two straight lines.
B
If the two lines are
perpendicular to each other,
then the angle between
the two lines is 90.
P
S
R
Q
Angle between Two Intersecting Straight Lines
The figure shows two non-parallel straight lines AB
and CD. They intersect each other at a point E.
D
A

C
E

The acute angle  formed is
called the angle between
two straight lines.
B
How about ∠AED and ∠CEB?
Are they also called the angle
between two straight lines?
Angle between Two Intersecting Straight Lines
The figure shows two non-parallel straight lines AB
and CD. They intersect each other at a point E.
D
A

C
E

The acute angle  formed is
called the angle between
two straight lines.
B
No, these obtuse angles are NOT
considered as the angle between
the two straight lines.
Can you identify the angle between
the lines AC and AG?
H
∵
AC and AG intersect at A .
∴
The angle between the lines
AC and AG is ∠GAC .
E
G
F
D
A
C
B
Follow-up Question
The figure shows a cube ABCDEFGH.
Identify the angles between
(a)
the lines BG and GH,
E
H
G
F
(b) the lines AD and BD,
D
C
(c) the lines AC and CF.
A
B
(a) ∵ BG and GH intersect at G.
∴ The angle between the lines BG and GH is BGH.
Follow-up Question
The figure shows a cube ABCDEFGH.
Identify the angles between
(a)
the lines BG and GH,
E
H
G
F
(b) the lines AD and BD,
D
C
(c) the lines AC and CF.
A
B
(b) Join BD.
∵ AD and BD intersect at D.
∴ The angle between the lines AD and BD is ADB.
Follow-up Question
The figure shows a cube ABCDEFGH.
Identify the angles between
(a)
the lines BG and GH,
E
H
G
F
(b) the lines AD and BD,
D
C
(c) the lines AC and CF.
A
B
(c) Join AC and CF.
∵ AC and CF intersect at C.
∴ The angle between the lines AC and CF is ACF.
Angle Between a Straight Line and a Plane
In the figure, P is a point lying outside the plane .
Q is a point on the plane  such that PQ is perpendicular to
any straight line on  passing through Q (e.g. L1 and L2).
Then,
P
(i) PQ  plane 
L1
Q

L2
(ii) Q is the projection
of P on the plane .
Angle Between a Straight Line and a Plane
In the figure, a straight line AB intersects the plane  at the
point A. C is the projection of B on the plane .
Then,
(ii) BAC is the angle
between the line AB
and the plane .
B
C
A
(i) AC is the projection
of AB on the plane .

Can you identify the angle
between the line FD and
the plane ABCD?
H
∵
∴
BD is the projection of FD
on the plane ABCD.
The angle between the line FD
and the plane ABCD is ∠FDB .
E
G
F
D
A
C
B
Follow-up question
The figure shows a right triangular prism with
∠ABC = ∠DEF = 90. Identify the angles
between
(a) the line AF and the plane ADEB.
(b)
the line AF and the plane DEF.
A
C
B
D
F
(a)
Join AF.
E
∵ AE is the projection of AF on the plane ADEB.
∴ The angle between the line AF and the plane ADEB
is ∠FAE.
Follow-up question
The figure shows a right triangular prism with
∠ABC = ∠DEF = 90. Identify the angles
between
(a) the line AF and the plane ADEB.
(b)
the line AF and the plane DEF.
A
C
B
D
F
(b)
∵
DF is the projection of AF on the
plane DEF.
∴
The angle between the line AF and the plane DEF
is ∠AFD.
E
After identifying angles in three
Let’s see
solve
dimensions,
wehow
can to
find
theirthe
sizes
followingtrigonometric
problems.
by applying
knowledge.
H
E
F
4 cm
F
6 cm D
A
E
G
D
C
5 cm
C
10 cm
B
Angle between the
lines CH and CD = ?
A
12 cm
B
Angle between the line AF
and the plane ABCD = ?
The figure shows a rectangular block
ABCDHEFG. AB = 10 cm and
AE = 6 cm. Find the angle between
the lines CD and CH, correct to
1 decimal place.
H
G
E
F
6 cm D
A
C
10 cm
B
Step 1: Identify the required angle.
∵ CD and CH intersect at C.
∴ The angle between the lines CD and CH is DCH.
The figure shows a rectangular block
ABCDHEFG. AB = 10 cm and
AE = 6 cm. Find the angle between
the lines CD and CH, correct to
1 decimal place.
H
E
G
F
6 cm D
A
10 cm
Step 2: Analyse the information needed to find the
required angle.
Do you know the
lengths of any side
Yes, I know the lengths of △CDH?
of CD and DH.
C
B
The figure shows a rectangular block
ABCDHEFG. AB = 10 cm and
AE = 6 cm. Find the angle between
the lines CD and CH, correct to
1 decimal place.
H
E
G
F
6 cm D
A
10 cm
Step 2: Analyse the information needed to find the
required angle.
Then you can find
∠DCH by using
CD = AB = 10 cm and
trigonometric ratio.
DH = AE = 6 cm
C
B
The figure shows a rectangular block
ABCDHEFG. AB = 10 cm and
AE = 6 cm. Find the angle between
the lines CD and CH, correct to
1 decimal place.
Solution
H
E
G
F
6 cm D
A
C
10 cm
B
The angle between the lines CD and CH is ∠DCH.
DC = AB = 10 cm
DH = AE = 6 cm
Consider △CDH.
∴ The angle between the lines
DH
CD and CH is 31.0.
tan DCH 
DC
 6 cm
10 cm
DCH  31.0 (cor. to 1 d.p.)
In some cases, Pythagoras
theorem is also useful
in finding the angles.
The figure shows a right triangular prism
ABCDEF. AB = 12 cm, BC = 5 cm,
CF = 4 cm and ∠BCF = ∠ADE = 90.
Find the angle between the line AF and
the plane ABCD, correct to 3 significant
figures.
E
F
4 cm
D
C
5 cm
A
12 cm
Step 1: Identify the required angle.
∵ AC is the projection of AF on the plane ABCD.
∴ The angle between the line AF and the plane ABCD
is ∠FAC.
B
The figure shows a right triangular prism
E
F
ABCDEF. AB = 12 cm, BC = 5 cm,
4 cm
CF = 4 cm and ∠BCF = ∠ADE = 90.
D
Find the angle between the line AF and
C
the plane ABCD, correct to 3 significant
5 cm
A
figures.
12 cm B
Step 2: Analyse the information needed to find the
required angle.
In △ACF,
CF = 4 cm
AC = ?
AF = ?
Consider
we can
Then△ABC,
we have
use
Pythagoras
theorem
enough
information
totofind
length
findthe
∠FAC.
of AC.
The figure shows a right triangular prism
ABCDEF. AB = 12 cm, BC = 5 cm,
CF = 4 cm and ∠BCF = ∠ADE = 90.
Find the angle between the line AF and
the plane ABCD, correct to 3 significant
figures.
Solution
E
F
4 cm
D
C
5 cm
A
12 cm
B
The required angle is FAC.
Consider △ACF.
CF
Consider △ABC.
tan FAC 
AC
2
2

◄
Pyth.
theorem
AC
AB + BC
4 cm

13 cm
 122 + 52 cm
FAC  17.1
 13 cm
(cor. to 3 sig. fig.)
∴ The angle between the line AF and the plane ABCD is 17.1.
Follow-up question
H
G
The figure shows a rectangular block
ABCDHEFG. AB = 8 cm, BC = 6 cm
E
F
and AE = 7 cm. Find the angle between
7 cm D
the line CE and the plane ABCD,
C
correct to 3 significant figures.
6 cm
A 8 cm B
∵ CA is the projection of CE on the plane ABCD.
∴ The angle between the line CE and the plane ABCD
is ∠ECA.
Consider △ABC.
AC 

AB2 + BC2 ◄ Pyth. theorem
82 + 62 cm
 10 cm
Follow-up question
H
G
The figure shows a rectangular block
ABCDHEFG. AB = 8 cm, BC = 6 cm
E
F
and AE = 7 cm. Find the angle between
7 cm D
the line CE and the plane ABCD,
C
correct to 3 significant figures.
6 cm
A 8 cm B
Consider △ACE.
AE
tan ECA 
AC
 7 cm
10 cm
ECA  35.0 (cor. to 3 sig. fig.)
∴ The angle between the line CE and the plane ABCD is 35.0.
Angle Between Two Intersecting Planes
It is given that plane 2 intersects plane 1
at a straight line AB.
A
2
B
1
AB is the line of
intersection of the
two planes.
Angle Between Two Intersecting Planes
If PX is a line on plane 1 such that PX  AB and
PY is a line on plane 2 such that PY  AB, then
X
A

P
2
P is a point on AB.
1
Y
B The acute angle  between PX
and PY is the angle between
the planes 1 and 2.
Can
younotice
identify
theDH
angle
between
Do
Therefore,
you
the that
required
⊥angle
AD
the
ADHE
and AFGD?
and
can planes
also
DG ⊥beAD?
∠HDG.
H
AD is the line of intersection of
the planes ADHE and AFGD.
E
G
F
D
∵
AE ⊥ AD and FA ⊥ AD
∴
A
The angle between the planes
ADHE and AFGD is ∠EAF. (or ∠HDG)
C
B
If AE = 14 cm and AF = 20 cm,
can you find the angle between
the planes ADHE and AFGD,
correct to 3 significant figures?
The required angle is ∠EAF.
Consider △AEF.
H
E
G
F
20 cm
AE
14 cm
cos EAF 
AF
D
C
 14 cm
B
A
20 cm
EAF  45.6 (cor. to 3 sig. fig.)
∴ The angle between the planes ADHE and AFGD is 45.6.
Follow-up question
E
F
The figure shows a triangular prism ABCDEF,
where ABCD, CFED and ABFE are rectangles.
It is given that AE = 12 cm, BC = 10 cm and 12 cm
C
D
∠BCF = 90.
10 cm
(a) Identify the angle between the
A
B
planes BEF and CDEF.
(b) Find the angle mentioned in (a), correct to 3 significant figures.
(a)
EF is the line of intersection of the planes BEF
and CDEF.
∵ BF ⊥ EF
and CF ⊥ EF
∴ The angle between the planes BEF and CDEF
is BFC.
Follow-up question
E
F
The figure shows a triangular prism ABCDEF,
where ABCD, CFED and ABFE are rectangles.
It is given that AE = 12 cm, BC = 10 cm and 12 cm
C
D
∠BCF = 90.
10 cm
(a) Identify the angle between the
A
B
planes BEF and CDEF.
(b) Find the angle mentioned in (a), correct to 3 significant figures.
(b)
Consider △BCF.
BC
sin BFC 
BF
10 cm
BF = AE = 12 cm

12 cm
BFC  56.4 (cor. to 3 sig. fig.)
∴ The angle mentioned in (a) (i.e. BFC) is 56.4.
Line of Greatest Slope
The figure shows an inclined road with two paths
AX and BX.
X
Let
us compare
I will
choose
the
of the
theslopes
path that
is
paths
AXsteep.
and BX.
less
Y
A
B
Which path do you
choose to reach X?
Line of Greatest Slope
X
Consider BXY.
XY
Slope of BX =
BY
Consider AXY.
Y
X
XY Slope of line
segment
Slope of AX =
AY = vertical distance
So,
I let
willus
choose
the
Now,
Here,
we
can
study
see how
that
horizontal
distance
AY < BY
∵
path
to
reach
X.
to
different
find BX
thestraight
line
with
lines
the
1
1
<
greatest
on an inclined
slope on
road
an
∴
BY AY
have
inclined
different
plane.
slopes.
XY XY
<
∴
BY
AY
i.e. Slope of BX < slope of AX
A
B
Line of Greatest Slope
The figure shows an inclined plane ABEF.
It intersects the horizontal plane ABCD at AB.
If PX  AB, the slope of PX is the greatest.
X
PX is the line of
greatest slope of
the inclined plane
ABEF.
P
The figure shows a hillside ABCD
sloping to the horizontal ground ABEF.
ABCD is a rectangle. It is given that
CE = 4 cm and BE = 15 cm and
∠BEC = 90.
A
D
C
4 cm
F
E
15 cm
B
Find the inclination of the line of
greatest slope of the hillside ABCD.
(Give your answer correct to
3 significant figures.)
The figure shows a hillside ABCD
sloping to the horizontal ground ABEF.
ABCD is a rectangle. It is given that
CE = 4 cm and BE = 15 cm and
∠BEC = 90.
A
D
C
4 cm
F
E
15 cm
B
∵ AB is the line of intersection of the inclined
plane ABEF and the horizontal plane ABCD,
and BC ⊥ AB.
∴ BC is the line of greatest slope of the
hillside ABCD.
The figure shows a hillside ABCD
sloping to the horizontal ground ABEF.
ABCD is a rectangle. It is given that
CE = 4 cm and BE = 15 cm and
∠BEC = 90.
A
D
C
4 cm
F
E
15 cm
B
Consider △BCE.
CE
tan CBE 
BE
4 cm

15 cm
CBE  14.9 (cor. to 3 sig. fig.)
∴ The inclination of the line of greatest slope of the
hillside ABCD is 14.9.
Follow-up question
The figure shows a rectangular board
ABCD sloping to a horizontal plane.
E and F are the projections of the points
C and D on the horizontal plane
respectively. It is given that CE = 5 cm, A
AB = 18 cm and BAC = 38.
D
C
5 cm
E
F
38
18 cm
B
(a) Find the length of the line of greatest slope of the
plane ABCD.
(b) Find the inclination of the line of greatest slope of
the plane ABCD.
(Give your answers correct to 3 significant figures.)
C
D
5 cm
E
F
A
38
18 cm
B
(a) BC is the line of greatest slope of the plane ABCD.
Consider △ABC.
BC
tan BAC 
AB
BC

tan 38
18 cm
BC  18 tan 38 cm
 14.1 cm (cor. to 3 sig. fig.)
∴ The length of the line of greatest slope is 14.1 cm.
D
BC  18 tan 38 cm
C
5 cm
E
F
A
38
18 cm
B
(b) ∠CBE is the inclination of the line of greatest slope of the
plane ABCD.
Consider △BCE.
sin∠CBE  CE
BC

5 cm
18 tan 38 cm
∠CBE  20.8 (cor. to 3 sig. fig.)
∴ The inclination of the line of greatest slope of the
plane ABCD is 20.8.
Distance between a Point and a Line
Consider a straight line L and a point P not lying on L.
If Q is a point on L such that PQ is
perpendicular to the line L, then,
P
PQ is the distance between
the point P and the line L.
Note: PQ is also the shortest distance between
the point P and the line L.
L
Q
Example:
The figure shows a cuboid ABCDEFGH.
It is given that AB = 9 cm, BC = 8 cm
F
and AF = 7 cm.
7 cm
A
E
H
G
D
C
8 cm
9 cm
The distance between the point A and the line BG
= 9 cm
The distance between the point B and the line CD
= 8 cm
The distance between the point F and the line AB
= 7 cm
B
Distance between a Point and a Plane
Consider a point R not lying on the plane , and its
projection, say point S on the plane .
Then, RS is perpendicular to the plane ,
and
RS is the distance between
the point R and the plane .
R

S
Note: RS is also the shortest distance between
point R and the plane .
D
The figure shows a rectangular board
ABCD inclining at 25 to the horizontal
plane. E and F are the projections of the
points C and D on the horizontal plane
respectively. It is given that BC = 15 cm.
C
15 cm
E
25
F
A
Find the distance between the
point B and the plane CDFE.
(Give your answer correct to
3 significant figures.)
B
D
The figure shows a rectangular board
ABCD inclining at 25 to the horizontal
plane. E and F are the projections of the
points C and D on the horizontal plane
respectively. It is given that BC = 15 cm.
C
15 cm
E
25
F
A
B
∵ E is the projection of B on the plane CDFE.
∴ BE is the distance between the point B and the plane CDFE.
Consider △BCE.
BE
cos CBE 
BC
BE
cos 25 
15 cm
BE  13.6 cm (cor. to 3 sig. fig.)
∴ The distance between the point B and the plane CDFE is 13.6 cm.
Follow-up question
The figure shows a rectangular block
ABCDEFGH. It is given that EH = 30 cm and
∠HFG = 20. Find the distances between
F
(a) the point G and the line EH,
(b) the point G and the plane ACHF.
(Give your answers correct to 3 significant
figures.)
A
(a)
30 cm
E
20
H
G
C
D
B
∵ GH ⊥ EH
∴ GH is the distance between the point G and the line EH.
Consider △FGH.
GH
tan HFG 
FG
GH
tan 20 
30 cm  FG = EH = 30 cm
GH  30 tan 20 cm
 10.9 cm (cor. to 3 sig. fig.)
∴ The distance between the point G
and the line EH is 10.9 cm.
E
20
F
(b) Let Q be the projection of point G on
the plane ACHF.
A
Then GQ is the distance between the
point G and the plane ACHF.
Consider △FQG.
QG
sin GFQ 
FG
sin 20  QG
30 cm
QG  30 sin 20 cm
 10.3 cm (cor. to 3 sig. fig.)
∴ The distance between the point G and the
plane ACHF is 10.3 cm.
30 cm
Q
H
G
C
D
B