Transcript Book 5 Chapter 16 Trigonometry (3)

16 Trigonometry (3)
Case Study
16.1 Applications in Two-dimensional Problems
16.2 Basic Terminology in Three-dimensional Figures
16.3 Applications in Three-dimensional Problems
Chapter Summary
Case Study
How can we walk up the hill in a
relatively more comfortable way?
It is more comfortable to walk up
an inclined road along a zigzag
path, let me explain it to you.
In order to explain the above question, we can use a prism to illustrate
the situation such that AB is the top of the inclined road and PQ is the
horizontal ground level.
As shown in the figure, M and N are the mid-points of BQ and DQ
respectively.
Although PMA is longer than PA, PM and MA are less steep than PA,
and hence they are more comfortable to walk up.
P. 2
16.1 Applications in Two-dimensional
Problems
A. Angle of Elevation and Angle of Depression
When we observe an object above us, the angle q between
our line of sight and the horizontal is called the angle of
elevation.
When we observe an object below us,
the angle f between the line of sight
and the horizontal is called the angle
of depression.
These two angles are important in solving practical trigonometric
problems.
P. 3
16.1 Applications in Two-dimensional
Problems
A. Angle of Elevation and Angle of Depression
Example 16.1T
In the figure, TB is a flag. The angles of elevation from a
point A to the top T and the base B of the flag are 35 and 20 respectively.
If the flag is 3 m long, find the distance between A and T.
Solution:
TAB  35  20
 15
TBA  90  20 (ext.  of )
 110
By sine formula,
AT
3m

sin 110 sin15
3 sin 110
AT 
m
sin 15
 10.9 m (cor. to 3 sig. fig.)
P. 4
16.1 Applications in Two-dimensional
Problems
A. Angle of Elevation and Angle of Depression
Example 16.2T
Fanny looks down from a platform at one end of a swimming
pool. There is a boy A at the far end of the pool and another boy B
in the pool between A and the platform. The boys are 25 m apart
and in the same lane. The angles of depression of boy A and boy B
from Fanny are 6 and 14 respectively.
(a) Find the height of the platform if Fanny’s eyes
are 1.7 m above the platform.
Solution:
ACB  8
CAB  6 (alt. s, // lines)
CBD  14 (alt. s, // lines)
(a) By sine formula,
BC
25 m

sin 6 sin 8
BC  18.7767 m
In CBD,
CD
18.7767 m
CD  4.5425 m
 The height of the platform
 (4.5425  1.7) m
 3 m (cor. to the nearest m)
sin 14 
P. 5
16.1 Applications in Two-dimensional
Problems
A. Angle of Elevation and Angle of Depression
Example 16.2T
Fanny looks down from a platform at one end of a swimming
pool. There is a boy A at the far end of the pool and another boy B
in the pool between A and the platform. The boys are 25 m apart
and in the same lane. The angles of depression of boy A and boy B
from Fanny are 6 and 14 respectively.
(a) Find the height of the platform if Fanny’s eyes
are 1.7 m above the platform.
(b) How far is boy B from the near end of the pool?
(Give the answers correct to the nearest m.)
Solution:
CBD  14 (alt. s, // lines), BC  18.7767 m
(b) In CBD,
BD
 BD  18.2190 m
cos14 
18.7767 m
 18 m (cor. to the nearest m)
 Boy B is 18 m from the near end of the pool.
P. 6
16.1 Applications in Two-dimensional
Problems
B. Bearing
In junior forms, we learnt how to use a compass bearing or
a true bearing to indicate the direction of an object from a
given point.
Compass bearing is also known as reduced bearing, and true bearing is
also known as whole circle bearing.
When using a compass bearing, directions are measured from the north (N)
or the south (S), thus the bearing is represented in the form:
Nq E, Nq W, Sq E or Sq W, where 0  q  90.
Notes:
If q  0 or 90, we simply write it as N, E, S or W.
When using a true bearing, all directions are measured from the north
in a clockwise direction.
The bearing is expressed in the form q, where 0  q  360 and written
in three digits such as 007, 056 or 198.
P. 7
16.1 Applications in Two-dimensional
Problems
B. Bearing
Example 16.3T
Eric and Frank are cycling away from P.
Eric is cycling in the direction 150 with a speed of 12 m/s and Frank is
cycling in the direction 220 with a speed of 10 m/s. After five minutes,
they stop and take a rest.
(a) What is the distance between them now?
(Give the answers correct to 1 decimal place.)
Solution:
(a) PE  (12  60  5) m  3600 m
PF  (10  60  5) m  3000 m
FPE  220  150  70
By cosine formula,
3000 m
FE  36002  30002  2(3600)(3000) cos 70 m
 14 572 364.9 m
 3817.3767 m
 3817.4 m (cor. to 1 d. p.)
P. 8
3817.3767 m
3600 m
16.1 Applications in Two-dimensional
Problems
B. Bearing
Example 16.3T
Eric and Frank are cycling away from P.
Eric is cycling in the direction 150 with a speed of 12 m/s and Frank is
cycling in the direction 220 with a speed of 10 m/s. After five minutes,
they stop and take a rest.
(a) What is the distance between them now?
(b) Find the true bearing of Frank from Eric.
(Give the answers correct to 1 decimal place.)
Solution:
(b) a  180  150  30, b  a  30 (alt. s, // lines)
By sine formula,
3000 m
PF
PE

sin E sin P
3817.3767 m
3000 sin 70
sin E 
 0.7385
True bearing
3817.3767
 360  47.6026  30
E  47.6026
 282.4 (cor. to 1 d. p.)
P. 9
3600 m
16.1 Applications in Two-dimensional
Problems
B. Bearing
Example 16.3T
Eric and Frank are cycling away from P.
Eric is cycling in the direction 150 with a speed of 12 m/s and Frank is
cycling in the direction 220 with a speed of 10 m/s. After five minutes,
they stop and take a rest.
(a) What is the distance between them now?
(b) Find the true bearing of Frank from Eric.
(c) After having a rest, if they cycle towards each other
at the same speed as before, how long does it take
for them to meet?
(Give the answers correct to 1 decimal place.)
Solution:
3000 m
3817.3767 m
(12  10) s
 2 min 53.5 s (or 2.9 min ) (cor. to 1 d. p.)
(c) Time taken 
P. 10
3817.3767 m
3600 m
16.2 Basic Terminology in Threedimensional Figures
A. Terms and Definitions
1. Angle between Two Straight Lines
The figure shows two intersecting straight lines
lying on the same plane.
The acute angle q is called the angle between the
two straight lines AB and CD.
In 3-D Figures, we can also identify the angle between two straight lines.
For example, the angle between BH and FH is BHF.
P. 11
16.2 Basic Terminology in Threedimensional Figures
A. Terms and Definitions
2. Angle between a Straight Line and a Plane
When a line is inclined on a plane, we can get
a projection of the line on the plane.
For example, when a javelin TP hits the ground,
the line AP is the projection of TP on the ground.
In three-dimensional space, the angle between a
straight line and a plane is the acute angle between
the straight line and its projection on the plane.
For example,
the projection of the line AG on AEHD is AH.
the angles between the line AG and AEHD is GAH.
Remark:
If the line is perpendicular to the plane, then the
projection of the line on the plane is only a point.
P. 12
16.2 Basic Terminology in Threedimensional Figures
A. Terms and Definitions
3. Angle between Two Planes
Consider the following two situations.
(a) A wooden door is opened.
(b) A greeting card is standing on a table.
In the above two cases, we observe that there are two planes
intersecting with each other.
P. 13
16.2 Basic Terminology in Threedimensional Figures
A. Terms and Definitions
When two planes intersect, they meet at a straight line
which is called the line of intersection.
In the figure, a and b are two planes while AQB is
the line of intersection.
PS and RT are lines on the planes a and b respectively
such that PQ ^ AB and RQ ^ AB.
The angle q between the lines PQ and RQ is called
the angle between planes a and b.
Remarks:
1. Actually, the angle between
two intersecting planes can be
acute or obtuse.
2. Usually, we do not consider the reflex angle as the angle between
two intersecting planes.
P. 14
16.2 Basic Terminology in Threedimensional Figures
A. Terms and Definitions
The figure shows a rectangular block.
 For planes ABFE and BCHE:
 Line of intersection:
_____________
BC
 Angle between 2 planes: _____________
ABE / DCH
 For planes CDEF and EFGH:
 Line of intersection:
_____________
EF
 Angle between 2 planes: _____________
CFG / DEH
The figure shows a right pyramid with a square base.
 For planes VCD and ABCD:
 Line of intersection:
_____________
CD
 Angle between 2 planes: _____________
VPQ
 For planes VBC and VCD:
 Line of intersection:
_____________
VC
 Angle between 2 planes: _____________
BND
P. 15
16.2 Basic Terminology in Threedimensional Figures
A. Terms and Definitions
4. Distance between a Point and a Straight Line
Consider a rectangular pyramid.
The distance between the point B and the line VC is
the perpendicular distance between B and VC, that is,
the length of BE.
5. Distance between a Point and a Plane
The distance between a point and a plane is the distance between
the point and its projection on the plane, that is, the perpendicular
distance between the point and the plane.
As shown in the figure, PQ is the distance between
point P and the plane.
P. 16
16.2 Basic Terminology in Threedimensional Figures
A. Terms and Definitions
Example 16.4T
The figure shows a cuboid. AB  3 cm, AD  6 cm and BF  4 cm.
(a) Find the length of AG and express the answer in surd form.
(b) Find the angle between the lines AG and AF.
(Give the answer correct to 3 significant figures.)
Solution:
(a) In EFG,
EG 2  EF 2  FG 2
EG  32  62 cm
 45 cm
In AEG,
AG 2  AE 2  EG 2
(Pyth. theorem)
(Pyth. theorem)
AG  4 2  ( 45) 2 cm
 61 cm
(b) FAG is the angle between
the lines AG and AF.
In AFG,
6
FG

sin FAG 
61
AG
FAG  50.2
(cor. to 3 sig. fig.)
 The angle between the lines
AG and AF is 50.2.
P. 17
16.2 Basic Terminology in Threedimensional Figures
A. Terms and Definitions
Example 16.5T
The figure shows a regular rectangular pyramid with base
12 cm  10 cm and slant height 15 cm. Suppose P is the mid-point of CD.
(a) Find VP and BP and give the answers in surd form if necessary.
Solution:
(a) In VPD,
VD 2  VP 2  PD 2
152  VP 2  52
VP  200 cm
 10 2 cm
(Pyth. theorem)
In BCP,
BP 2  BC 2  CP 2 (Pyth. theorem)
BP  122  52 cm
 169 cm
 13 cm
P. 18
16.2 Basic Terminology in Threedimensional Figures
A. Terms and Definitions
Example 16.5T
The figure shows a regular rectangular pyramid with base
12 cm  10 cm and slant height 15 cm. Suppose P is the mid-point of CD.
(a) Find VP and BP and give the answers in surd form if necessary.
(b) Find the angle between
(i) lines VB and VD, (ii) lines VP and BP.
(Give the answers correct to 3 significant figures.)
Solution:
(b) (i) BVD is the angle between lines VB and VD.
In BCD,
In BVD, by cosine formula,
152  152  ( 244 ) 2
BD 2  BC 2  CD 2
cos BVD 
(Pyth. theorem)
2(15)(15)
206
BD  122  102 cm

450
 244 cm
BVD  62.8 (cor. to 3 sig. fig.)
 The angle between lines VB and VD is 62.8.
P. 19
16.2 Basic Terminology in Threedimensional Figures
A. Terms and Definitions
Example 16.5T
The figure shows a regular rectangular pyramid with base
12 cm  10 cm and slant height 15 cm. Suppose P is the mid-point of CD.
(a) Find VP and BP and give the answers in surd form if necessary.
(b) Find the angle between
(i) lines VB and VD, (ii) lines VP and BP.
(Give the answers correct to 3 significant figures.)
Solution:
(b) (ii) BPV is the angle between lines VP and BP.
In BVP, by cosine formula,
132  (10 2 ) 2  152
cos BPV 
2(13)(10 2 )
144

260 2
BPV  66.9 (cor. to 3 sig. fig.)
 The angle between lines VP and BP is 66.9.
P. 20
16.2 Basic Terminology in Threedimensional Figures
A. Terms and Definitions
Example 16.6T
The figure shows a wedge with rectangular planes ABCD,
EFDA and EBCF. AB  10 cm, BC  16 cm, DCF  35 and DF ^ CF.
(a) Find the lengths of BD and DF.
(b) Find the angle between line BD and plane EBCF.
(Give the answers correct to 3 significant figures.)
Solution:
(a) In ABD,
In CDF,
BD 2  AB 2  AD 2 (Pyth. theorem) CD  AB  10 cm
DF
BD  102  162 cm
sin 35 
10 cm
 356 cm
 18.9 cm (cor. to 3 sig. fig.)
DF  5.74 cm (cor. to 3 sig. fig.)
(b) Since BF is the projection of
DF 10 sin 35
sin DBF 

BD on plane EBCF, DBF
BD
356
is the required angle.
DBF  17.7 (cor. to 3 sig. fig.)
 The angle between line BD and plane EBCF is 17.7.
P. 21
16.2 Basic Terminology in Threedimensional Figures
A. Terms and Definitions
Example 16.7T
The figure shows a right-angled triangular prism with ABCD,
AEFD and BCFE as rectangular faces. P is the mid-point of BC. DF  6 cm,
FC  8 cm and AD  15 cm. Find the angle between
(a) the line AC and the plane BCFE;
(b) the line AP and the plane BCFE.
(Give the answers correct to 3 significant figures.)
Solution:
(a) Since EC is the projection of AC on the plane BCFE, ACE is
the required angle.
In CEF,
In ACE,
AE 6
CE 2  EF 2  CF 2 (Pyth. theorem)
tan ACE 

CE 17
CE  152  82 cm
ACE  19.4
 17 cm
(cor. to 3 sig. fig.)
 The angle between the line AC and the plane BCFE is 19.4.
P. 22
16.2 Basic Terminology in Threedimensional Figures
A. Terms and Definitions
Example 16.7T
The figure shows a right-angled triangular prism with ABCD,
AEFD and BCFE as rectangular faces. P is the mid-point of BC. DF  6 cm,
FC  8 cm and AD  15 cm. Find the angle between
(a) the line AC and the plane BCFE;
(b) the line AP and the plane BCFE.
(Give the answers correct to 3 significant figures.)
Solution:
(b) Since EP is the projection of AP on the plane BCFE, APE is
the required angle.
In BEP, BP  7.5 cm.
In AEP,
AE
6
tan APE 

EP 2  BE 2  BP 2 (Pyth. theorem)
EP
120.25
EP  82  7.52 cm
APE  28.7
 120.25 cm
(cor. to 3 sig. fig.)
 The angle between the line AP and the plane BCFE is 28.7.
P. 23
16.2 Basic Terminology in Threedimensional Figures
A. Terms and Definitions
Example 16.8T
The figure shows a pyramid with a right-angled triangular base.
AB  AC  4 cm, VA  5 cm and VAB  VAC  90.
(a) Find the length of AM where M is the mid-point of BC in surd form.
(b) Find the angle between the planes ABC and VBC.
(Give the answer correct to 3 significant figures.)
Solution:
(a) ABC  ACB (base s, isos. )
180  90
( sum of )

2
 45
In ABM,
AM
4 cm
AM  2 2 cm
sin 45 
(b) VMA is the angle between the planes ABC and VBC.
5
 VMA  60.5 (cor. to 3 sig. fig.)
In AVM, tan VMA 
2 2
 The angle between the planes ABC and VBC is 60.5.
P. 24
16.2 Basic Terminology in Threedimensional Figures
B. Lines of Greatest Slope
In the figure, the inclined plane ABCD intersects the horizontal
plane ABEF at the line AB.
Three lines XY, PQ and ST are drawn on the
inclined plane with PQ perpendicular to AB.
Let a, b and g be the angles that XY, PQ and
ST make with the horizontal plane respectively.
If we compare the three angles, we find that b > a and b > g.
In fact, PQ makes the largest angle with the horizontal plane and it is
called the line of greatest slope of the inclined plane.
Notes:
There are infinitely many lines of greatest slope
on a given inclined plane, such as l1, l2, and l3
(that are parallel to the line PQ) in the figure.
P. 25
16.2 Basic Terminology in Threedimensional Figures
B. Lines of Greatest Slope
Example 16.9T
The figure shows a right-angled triangular prism with ABCD,
BCEF and AFED as rectangles. M and N are the mid-points of BC and AD
respectively. If EC  5 cm, DE  2 cm and DA  5DE, find
(a) the angle between the BN and plane BCEF,
(b) the angle between line NC and plane BCEF,
(c) the inclination of the line of greatest slope of plane ABCD.
(Give the answers correct to 3 significant figures.)
Solution:
BF  5 cm and AD  FE  10 cm
Let P be the mid-point of EF.
FP  5 cm and NP  2 cm
(a) NBP is the required angle.
In BFP,
BP  52  52 cm
 50 cm
(Pyth. theorem)
P
2
 0.2828
50
NBP  15.79
 The required angle is 15.8.
 tan NBP 
P. 26
16.2 Basic Terminology in Threedimensional Figures
B. Lines of Greatest Slope
Example 16.9T
The figure shows a right-angled triangular prism with ABCD,
BCEF and AFED as rectangles. M and N are the mid-points of BC and AD
respectively. If EC  5 cm, DE  2 cm and DA  5DE, find
(a) the angle between the BN and plane BCEF,
(b) the angle between line NC and plane BCEF,
(c) the inclination of the line of greatest slope of plane ABCD.
(Give the answers correct to 3 significant figures.)
Solution:
(b) NCP is the required angle.
Since CPN  BPN (SAS).
 NCP  NBP
 15.8 (cor. to 3 sig. fig.)
P
(c) Line of greatest slope: CD  DCE is the required angle.
2
tan DCE 
 DCE  21.8 (cor. to 3 sig. fig.)
5
P. 27
16.3 Applications in Three-dimensional
Problems
In this section, we shall further study some applications of
trigonometric formulas in three-dimensional figures, together
with bearings and angles of elevation and depression.
P. 28
16.3 Applications in Three-dimensional
Problems
Example 16.10T
The figure shows the plane of a hillside. E is due west of F and
C is due south of F. The inclination of the path CD is 8. DF  10 m and
AD  65 m. Eric runs directly from C to A.
(a) Find the inclination of his path with the ground EBCF.
(b) Find the compass bearing of his path from C.
(Give the answers correct to 3 significant figures.)
Solution:
(a) Since EC is the projection of AC on the ground EBCF, ACE is the
required angle.
In CDF,
In ACD, AC 2  AD 2  CD 2 (Pyth. theorem)
2
2
10 m
AC

65

71
.
853
m
sin 8 
 96.891 m
CD
In ACE,
10
10
CD 
m
sin ACE 
 ACE  5.92
sin 8
96.891
(cor. to 3 sig. fig.)
 71.853 m  The inclination of his path with the ground
EBCF is 5.92.
P. 29
16.3 Applications in Three-dimensional
Problems
Example 16.10T
The figure shows the plane of a hillside. E is due west of F and
C is due south of F. The inclination of the path CD is 8. DF  10 m and
AD  65 m. Eric runs directly from C to A.
(a) Find the inclination of his path with the ground EBCF.
(b) Find the compass bearing of his path from C.
(Give the answers correct to 3 significant figures.)
Solution:
(b) The projection of AC on the ground EBCF is EC.
In CDF,
10 m
tan8 
CF
10
CF 
m
tan 8
 71.1537 m
In CEF, EF  AD  65 m.
65
tan ECF 
71.1537
ECF  42.4122
 42.4 (cor. to 3 sig. fig.)
 The compass bearing of Eric’s path
from C is N42.4W.
P. 30
16.3 Applications in Three-dimensional
Problems
Example 16.11T
A lighthouse VA with height 90 m stands on the same plane as
two ships P and Q. The bearings of the lighthouse from P and Q are
N50 E and N65 W respectively. The angle of elevation of V from P
is 15 and the distance between P and Q is 800 m.
(a) Find the distance between P and A.
(b) Find the distance between Q and A.
(Give the answers correct to 3 significant figures.)
Solution:
(a) In VAP,
(b) PAQ  50  65  115 (alt. s, // lines)
90 m
By cosine formula,
tan15 
PA
QP2  PA2  QA2  2( PA)(QA) cos PAQ
90 m
8002  335.88462  QA2
PA 
tan 15
 2(335.8846)(QA) cos115
 335.8846 m  QA2  283.9019(QA)  527181.5355  0
 336 m
QA  598 m or 882 m (rejected)
(cor. to 3 sig. fig.)
(cor. to 3 sig. fig.)
P. 31
16.3 Applications in Three-dimensional
Problems
Example 16.11T
A lighthouse VA with height 90 m stands on the same plane as
two ships P and Q. The bearings of the lighthouse from P and Q are
N50 E and N65 W respectively. The angle of elevation of V from P
is 15 and the distance between P and Q is 800 m.
(a) Find the distance between P and A.
(b) Find the distance between Q and A.
(c) Hence find the angle of elevation of V from Q.
(Give the answers correct to 3 significant figures.)
Solution:
(c) In VAQ,
90
597.8677
VQA  8.5607
 8.56 (cor. to 3 sig. fig.)
tan VQA 
 The angle of elevation of V from Q is 8.56.
P. 32
Chapter Summary
16.1 Applications in Two-dimensional Problems
1. Angles of elevation and depression
The angle between the line of sight of an object above
us and the horizontal is the angle of elevation.
The angle between the line of sight of an object below
us and the horizontal is the angle of depression.
2. (a) Compass bearing
All directions are measured from the north (N) or the south (S).
The bearing is expressed in the form Nq E, Nq W, Sq E or Sq W,
where 0  q  90.
(b) True bearing
All directions are measured from the north in a clockwise direction.
The bearing is expressed in the form q, where 0  q  360 and
written in three digits.
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Chapter Summary
16.2 Basic Terminology in Three-dimensional
Figures
1. Angle between Two Straight Lines
The angle between two intersecting straight lines is the
acute angle formed by the two straight lines lying on
the same plane.
2. Angle between a Straight Line and a Plane
The angle between a straight line and a plane is the
acute angle between the straight line and its projection
on the plane.
3. Angle between Two Planes
The angle between two planes is the angle between two
perpendiculars on the respective planes to the line of
intersection of the two planes.
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Chapter Summary
16.2 Basic Terminology in Three-dimensional
Figures
4. Distance between a Point and a Straight Line
The distance between a point and a straight line is the
perpendicular distance from the point to the line.
5. Distance between a Point and a Plane
The distance between a point and a plane is the
distance between the point and its projection on
the plane.
6. Lines of Greatest Slope
If PQ ^ AB, then PQ is called the line of greatest
slope of the inclined plane ABCD.
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Chapter Summary
16.3 Applications in Three-dimensional Problems
In three-dimensional figures, we can also find
(a) angles of elevation and depression, and
(b) bearing.
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