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Immaculate Heart of Mary College
S3 Remedial Class
Lesson 20: Applications of Trigonometry
Name: ________________________(
) Class: S3 (
) Date: __________
Basic Knowledge
Note:
In junior form, when you are solving a trigonometric problem, you must have a
right-angled triangle.
Application 1: The area of some plane figures
Example 1 Find the area of the triangle below.
A
Answer
tan 50 
BC 
8
BC
8
tan 50
The area of the triangle is:
1
8

8
2 tan 50
8 cm
50
B
C
 26.9cm 2 (corr. to 3 sig. fig.)
Example 2 Find the area of the triangle below.
Answer
We do not have right-angled triangle here. So
we add a line AD to solve this problem.
sin 30 
AD
7
1 AD

2
7
AD  3.5
The area of the triangle is:
1
 8  3.5  14cm 2
2
1
Example 3
Find the area of the parallelogram below.
Answer
We do not have right-angled triangle here. So
we add a line QT to solve this problem.
sin 60 
P
4 cm
2 3 cm
QT
2 3
Q
60
S
R
3 QT

2
2 3
3
2 3
2
QT  3
QT 
The area of the parallelogram is:
4  3  12cm 2
Example 4
Find the area of the trapezium below. (Express your answer in surd
form.)
Answer
We need 1 more right-angled triangle. So
we add a line EK to solve this problem.
tan 60 
8 3
GH
8 3
GH
1
GH 
8 3
3
3
GH  8
WLOG, KF  8
The area of the trapezium is:
1
 [15  (8  15  8)]  8 3  184 3cm 2
2
2
Exercise 1:
1. Find the area of each of the following triangles.
(a)
(b)
(c)
L
A
P
60
7 cm
12 cm
30
B
Q
60
M
R
2. Find the area of each of the following quadrilaterals.
(a)
(b)
P
10 cm
N
5 cm
C
(c)
Q
A
5 cm
Q
P
B
30
10 3 cm
10 cm
45
60
S
D
9 cm
C
S
R
R
Application 2: Gradient and Geographical Gradient
3
Example 5 The gradient of a slope is 2 in 5.
(a) Find the angle of inclination of the slope.
(b) What is the walking distance if the vertical rise is 40m?
(Correct your answers to 3 significant figures.)
Answer
(a) Let  be the angle of inclination of the slope
t a n 
2
5
  21.8
∴The angle of inclination of the slope is 21.8 .
(b) Let x be the walking distance of the slope
sin 21.8 
40
x
40
sin 21.8
x
x  108m (corr. to 3 sig. fig.)
Example 6
The figure shows a part of a map with the scale of 1 cm : 0.25 km. The
length of XY in the map is 1.8 cm.
(a) Find the actual distance of XY.
(b) Find the angle of inclination of XY.
550 m
600 m
650 m
Answer
(a) The actual distance of XY is:
1.8  0.25
 0.45km
 450m
X
Y
Scale 1 cm : 0.25 km
(b) Let  be the angle of inclination of XY
t a n 
6 5 0 5 5 0
450
tan  
100
450
  12.5 (corr. to 3 sig. fig.)
Exercise 2:
4
1. A car travels 3.2 km up a slope with the gradient of 1 : 15. Find the vertical rise of
the car.
2. The horizontal and vertical distances of a road are 420 m and 35 m respectively.
Find the gradient of the road.
1
3. A man walks 240 m down a path with the gradient of 8 . Find the horizontal and
vertical distances he walks.
4. The figure shows a map with the scale of 1 : 10 000. O denotes the location ofa
signpost, and OA and OB denote two paths to there. On the map, OA  2.8 cm and
OB  1.6 cm.
(a) Find the angle of inclination of OA.
(b) Find the angle of inclination of OB.
260 m
(c) Which path is flatter?
240 m
O
A
220 m
200 m
180 m
B
160 m
Scale 1 : 10 000
Application 3: Angle of elevation and depression
5
Example 7 The figure shows that a person in a boat looks at point A of a lighthouse
at the angle of elevation 19.5. If the distances of the eye level and point
A from the sea level are 2.4 m and 20 m respectively, find the horizontal
distance between the boat and point A. (Correct your answers to 3
significant figures.)
A
20 m
19.5
2.4 m
Answer
Let h be the horizontal distance between the boat and point A.
tan 19.5 
h
20  2.4
h
17.6
tan 19.5
h  49.7m
Exercise 3:
1. A person looks at the top Y of building B from the top X of building A at the angle
of elevation 28. If the height of building A is 40 m and these two buildings are
240 m apart, find the height of building B.
Y
X
28
40 m
A
B
240 m
2. A person looks at peak B from peak A at the angle of depression 7.4. It is given
that the heights of peaks A and B are 627 m and 508 m respectively. Find the
horizontal distance between the two peaks.
A
7.4
B
627 m
3. A
508 m
person sits
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at the front A of the upper deck of a bus and looks at the top B of a building. Given
that A and B are 3.5 m and 120 m from the ground respectively, and the front of the
bus is 164 m away from the building, find the angle of elevation to B from the eyes
of the person.
B
120 m

A
3.5 m
164 m
4. Two people look at point C on the ground from A and B of a building at the angles
of depression 42 and 30 respectively. If A is 14.2 m above B, find the height of A
from the ground.
A
42
14.2 m
B
30
C
Application 4: Bearing
We have 2 ways of indicating the bearing: compass bearing and true bearing.
Example 8 According to the figure, find
(a) the true bearing of Y from X.
(b) the compass bearing of X from Y.
N
Y
N
X
15
Answer
(a) The true bearing of Y from X is 075 .
(b) The compass bearing of X from Y is S 75W .
Example 9
In the figure, a ship sails 10 km from A to B. The bearing of B from A is
N45W. Then the ship sails 15 km to C from B at the direction N20W.
(a) Find the true bearing of C from A.
C
(b) Find the distance of AC.
N
15 km
20
45
N
7
B
10 km
A
Answer
As shown in the figure,
(a) We must find CAF .
BAD  90  45  45
sin 45 
AD
10
cos 45 
2 AD

2
10
2
 10
2
 7.0711
AD 
BD
10
2 BD

2
10
2
 10
2
 7.0711
BD 
CBE  90  20  70
sin 70 
CE
15
CE  15  sin 70
 14.0954
cos 70 
BE
15
BE  15  cos 70
 5.1303
CF  BE  DA  12.2 0 1 4
AF  CE  DB  21.1665
tan CAF 
CF
AF
CAF  30.0 (corr. to 3 sig. fig.)
So the true bearing of C from A is 330 .
(b) The distance of AC is:
AC  CF 2  AF 2 (Pyth.Theorem)
 24.4km
Exercise 4:
1. Jane walks 500 m due south and then 750 m due west. Find the compass bearing
of her final position from the starting point.
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2. Vivian walks 5 km from A at a bearing of N62W to B, and then walks 3 km at a
bearing of S28W to C. Find the true bearing of C from A.
N
B
N
5 km
3 km
62
28
A
C
3. The figure shows three locations P, Q and R on a map. P is at a bearing of N28W
from Q, Q is at a bearing of S62W from R, and P is at a bearing of S95W from
R. It is given that PQ  45 km.
(a) Find the distance between P and R.
(b) Find the distance between Q and R.
N
95
R
P
28
62
N
Q
4. The figure shows the locations of three lighthouses X, Y and Z. Ben sails 19 km in
a boat from lighthouse X to lighthouse Y at a bearing of 203, and then sails 22 km
to lighthouse Z at a bearing of 102.
(a) Find the distance between lighthouses X and Z.
(b) Find the true bearing of X from Z
N
X
203
19 km
N
Y
102
N
22 km
Z
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