F.3 Mathematics Supplementary Notes
Ch.10 Trigonometric Relations
Ch.11 Applications in Trigonometry 5/2006 P. 1
Chapter 10 Trigonometric Relations
Chapter 11 Applications in Trigonometry
Important Terms
adjacent side
hypotenuse
opposite side
trigonometric ratio
sine ratio
cosine ratio
Name:_______________( )
Class: F.3 ( )
tangent ratio
compass bearing
true bearing
angle of elevation
angle of depression
Revision Notes
1. The trigonometric ratios applied to a right-angled triangle
 ABC which is right-angled at C, we have :
For a
cos 
adjacent side of  AC
=
AB
hypotenuse
B
side of 
opposite side of BC
sin  

hypotenuse
AB
tan  
2.

C
adjacent
side of 
opposite side of
BC

adjacent side of  AC
The trigonometric ratios of the special angles 0, 30, 45, 60 and 90.
sin
cos
tan
3.
hypotenuse
opposite
0
Trigonometric Identities
(a)
tan  
sin 
cos 
(b) sin 2   cos 2   1
(c) sin   cos(90    )
(d) cos  sin( 90   )
1
(e) tan  
tan( 90    )
30
45
60
90
A
F.3 Mathematics Supplementary Notes
4.
Ch.10 Trigonometric Relations
Ch.11 Applications in Trigonometry 5/2006 P. 2
Bearings
Compass bearing
All directions are measured either from the north or
from the south. The angle used ranges from
0 to 90 .
N
D
35
A
True bearing
All directions are measured from the geographical
north in a clockwise direction. The angle used is
usually expressed in three digits which range from
000 to 360.
N
A
65
20
O
W
E
70
C
45
B
240
C
S
B
e.g. the compass bearing of
(i) A from O is
N65E
(ii) B from O is ________
(iii) C from O is ________
(iv) D from O is ________
5.
140
O
e.g. the true bearing of
(i) A from O is
020
(ii) B from O is _______
(iii) C from O is _______
Angle of elevation and angle of depression
A
line of sight
angle of elevation of A from O
O
horizontal
angle of depression of B from O
line of sight
B
Exercise Level 1
1. Referring to the given  ABC, find the values of
(a)
cos  
21
29
(b) cos  
C
20
29
21
20
α
β
B
(c)
sin  
(d) sin  
(e)
tan  
(f)
tan  
29
A
A
F.3 Mathematics Supplementary Notes
2.
Ch.10 Trigonometric Relations
Ch.11 Applications in Trigonometry 5/2006 P. 3
Find the values of the following. (Give your answers correct to 3 decimal places if necessary.)
(a) sin 20∘=
(b) cos55∘=
(c) sin 7.8∘=
(d) tan 24.38∘=
3. Find the values of  in the following.( Give your answers correct to the nearest 0.01∘if necessary.)
(a) sinθ= 0.37
(b) cos  = 0.2
θ= ______
(c)
tan  
1
7
4. Given that sin  
5.
6.
(d) cos 
3
4
3
, find the values of cos  and sin  . (Give your answers in surd form whenever possible.)
5
Find the values of the following (Give your answers in surd form whenever possible.)
(a)
sin 45 2 sin 30 

cos 60 
tan 45
(c)
2
1

0
cos 0
sin 90 0
(b) (sin 30   tan 45 )(tan 30   sin 60  )
(d)
sin 0 0  cos 90 0
tan 30 0 tan 60 0
(b)
cos 
 tan 
sin 
Simplify the following:
(a)
1
1
cos 2 
F.3 Mathematics Supplementary Notes
7.
Ch.10 Trigonometric Relations
Ch.11 Applications in Trigonometry 5/2006 P. 4
(c)
tan 19 tan 71
1  cos 2 32 
(d)
cos 2 58
(e)
sin( 90    ) cos  cos(90    ) sin 
(f)
3
sin 

0
tan( 90   ) sin( 900   )
(b)
1
 sin( 90   )  tan  cos(90   )
cos


Prove the following
(a)
cos sin 
sin 2 
(

)
1
sin  cos
tan 
Level II
8. Find the values of  in the following.( Give your answers correct to the nearest 0.01∘if necessary.)
2
2
(a)
tan   1  2
(b) cos 
(c)
cos  sin 30 
(d) 2 sin   2
(e)
2 2 cos  1
(f)
tan  tan 72   1
F.3 Mathematics Supplementary Notes
Ch.10 Trigonometric Relations
(g) 3 tan 2  3  0
(i)
9.
(h) cos 2  sin 36 
sin( 3  15 )  cos 51
Given that tan  
Ch.11 Applications in Trigonometry 5/2006 P. 5
1
2 2
(j)
tan 3 
1
tan( 30    )
, find the value of cos  + sin  . (Give your answers in surd form.)
10. Prove the following
(a)
cos
1  sin 
2


1  sin 
cos
cos
(b) cos 2 (90    )  (1  cos ) 2 = 2(1  cos  )
11. (a) Prove that (sin   cos ) 2  1  2 sin  cos .
(b) If sin   cos 
1
, find the value of sin  cos .
2
F.3 Mathematics Supplementary Notes
Ch.10 Trigonometric Relations
Ch.11 Applications in Trigonometry 5/2006 P. 6
12. Evaluate the following
(a)
1  tan 2 30 
tan 2 60 
(c)
sin 45 2 sin 30 

cos 60 
tan 45
(b)
(
1
1
 cos 60  )(
 sin 60  )

sin 30
cos 30 
(d)
cos 54 
tan 36 tan 54 
sin 54 
(a)
cos 2 
1
1  sin 
(b)
tan  sin  
(c)
sin 2   (sin 2  cos 4   sin 4  cos 2  )
(d)
2


13. Simplify
1
1
1
cos

1
cos 
1
1
1
cos
F.3 Mathematics Supplementary Notes
Ch.10 Trigonometric Relations
Ch.11 Applications in Trigonometry 5/2006 P. 7
14. In ABC, CD =15,∠ABC=45and ∠BAC = 75. AD is the perpendicular from A to BC. Find
(a) ACB ,
(b) AC, BD.
A
(Give the answers in surd form if necessary.)
75
B
45
C
D 15
15. In the figure ACB is right-angled at B. BD=CD=1,∠DAB=  and ∠CAD = .
(a) Express AB in terms of  .
C
(b) Express tan∠CAB in terms of  .
1
(c) If  =35, find  correct to the nearest degree.
D

1

A
B
16. In the figure, PQ is a flagpole and PR is a support wire. S is a point on the ground 5m away from R
such that the angle of elevation of P from S is 40. From R, the angle of elevation of P is 70. Find
(a) the length of PR,
P
(b) the length of PQ.
Q
70
40
S
R
5m
17. In the following figures, find the unknowns:
(Give your answers correct to 3 significant figures.)
(a)
A
20
B
C
16

12
D
F.3 Mathematics Supplementary Notes
Ch.10 Trigonometric Relations
Ch.11 Applications in Trigonometry 5/2006 P. 8
(b)
A
F
10cm
30cm
D
h
B
45∘
E
C
18. (a) Refer to the given figure, show that BD=


6  2 cm.
A
(b) Find the length of DE.
(c) Show that sin 15  
6 2
.
4
15 
2 cm
E
45 
30 
B
C
D
2 cm
19. The figure shows a triangular field PQR with PQ=800m and QR=600m.
The true bearings of P and R from Q are 045 and 315 respectively.
Find the area of the field.
R
P
N
600m
800m
45
Q
CEMC91Q
20. In the figure, A and B are the positions of two boats.
Find the bearing of B from A.
N
A
35
75
O
B
E
F.3 Mathematics Supplementary Notes
Ch.10 Trigonometric Relations
Ch.11 Applications in Trigonometry 5/2006 P. 9
CEMC94
21. The bearing of A from B is 075. What is the bearing of B from A?
A
CEMC95Q
N
22. In the figure, find the bearing of B from A.
30
O
B
CE97Q6
23. In the figure, the bearings of two Ship A and B from a lighthouse L are 020
N
and 110 respectively. B is 20 km and at a bearing of 140 from A. Find
(a) the distance of L from B,
A 140∘
(b) the bearing of L from B.
N
20∘
20 km
L
110∘
B
D
CEMC88Q17
24. In the figure, DA=h and ∠CDB=∠BDA=  . Find
AC
.
AB
 
h
A
B
C
F.3 Mathematics Supplementary Notes
Ch.10 Trigonometric Relations
Ch.11 Applications in Trigonometry 5/2006 P. 10
25. The figure shows a part of a contour map of scale 1:20 000. AB represents a straight path
connecting points A and B on different contour lines. Suppose the length of AB on the map is 5 cm.
(a) Find the real length of the straight path. (Give the answer correct to the nearest m.)
(b) Find the inclination of the straight path. (Give the answer correct to the nearest 0.1.)
B
400 m
350 m
300 m
250 m
Scale 1:20 000
A
F.3 Mathematics Supplementary Notes
Ch.10 Trigonometric Relations
Ch.11 Applications in Trigonometry 5/2006 P. 11
************************************OPTIONAL***************************************
Level III
1. Prove that in any triangle  ABC,
(a)
2.
sin
A
BC
 cos

2
 2 
A 
BC

(b)  tan    tan
 1
2 
2 

ABCD is a square. M and N are mid-points of AB and AD respectively. Find the value of  .
(Give your answers correct to the nearest 0.01∘) (Ans: 36.9∘)
M
A
B
N

D
C
3.
In  ABC, AB = c, BC = a and ∠ABC =  .
(a) Express h in terms of c and  .
(b) Express the area of  ABC in terms of a, c and  .
A
c
B
h

C
a
4.
ABCDEF is a regular hexagon inscribed in a circle of radius 20cm. Find
(a) ∠AOB,
F
A
(b) area of ABCDEF,
(c) perimeter of ABCDEF.
(Ans: 60∘, 600 3 cm2 , 120cm)
B
20cm
C
E
O
D
F.3 Mathematics Supplementary Notes
5.
Ch.10 Trigonometric Relations
Ch.11 Applications in Trigonometry 5/2006 P. 12
In the figure, ABC is an equilateral triangle and the radii of the three circles are each equal to 1.
C
Find the perimeter of the triangle.
(Ans.: 6  6 3 )
B
A
2t
, express cos and tan  in terms of t .
1 t2
6.
If sin  
7.
Find the values of the following.
(a)
(Ans: cos  
2t
1 t2
, tan  
)
2
1 t2
1 t
tan 41  tan 42     tan 49 
(Ans: 1 )
89
(Ans: )
2
(b) sin 2 1 + sin 2 2  + sin 2 3 ……+ sin 2 89 
8.
In the figure,  ABC is an isosceles triangle. AB = AC =1 ,∠BAC =36∘and BD bisects∠ABC.
(a) Prove  ABC ~  BCD.
A
(b) Let BC = x. Show that x 2  x  1  0 .
5 1
is a root of x 2  x  1  0 .
2
Hence find the value of cos 36 . (Give your answers in surd form)
(c) Show that
(Ans.:
36∘
5 1
)
4
1
B
D
x
C
F.3 Mathematics Supplementary Notes
Ch.10 Trigonometric Relations
Ch.11 Applications in Trigonometry 5/2006 P. 13
數學課外閱讀 :
內容簡介 :
聽到三角，絕大多數人的腦海可能馬上就閃過一大堆  、  、sin、tan、csc、半角公式、
倍角公式、和化積、積化和、......，管它實不實用，
「卯」起來背就對了，還可以「說」些什
麼呢？如果你真的這麼認為，以為三角學只是一個又一個函數與公式，那可大錯特錯了！
學校裡，從沒有人告訴你三角的歷史和生命，也沒有人告訴你三角和人類文明如何緊緊
相扣；從古埃及的金字塔、砲彈的射程、精確地圖的誕生，到天文測量，全都離不開三角。
其實，三角學與現實世界是密不可分的。
為了傳達這個觀念，本書作者毛爾教授捨棄教科書式的枯燥寫法，
「毛」起來說了許多關
於三角的小故事：透過這本書，你將目睹三角學如何從埃及金字塔裡萌芽、從古希臘乃至中
世紀天文學家的手中茁壯，你也將認識許許多多偉大的數學家，跟隨他們的腳步，踏進有趣
的三角殿堂，一探平面三角學背後的無窮樂趣。
【毛起來說三角】
(Trigonometric Delights)
毛爾 著 胡守仁 譯
出 資料來源:
http://www.cp1897.com.hk/BookInfo?BookId=9576217326&SectionId=10&AllId=0&Action=50
版社: 天下文化
F.3 Mathematics Supplementary Notes
Ch.10 Trigonometric Relations
Ch.11 Applications in Trigonometry 5/2006 P. 14
F.3 Mathematics Supplementary Notes
Ch.10 Trigonometric Relations
Ch.11 Applications in Trigonometry 5/2006 P. 15
F.3 Mathematics Supplementary Notes
Ch.10 Trigonometric Relations
Ch.11 Applications in Trigonometry 5/2006 P. 16