The document is about trigonometry and includes: 1. An introduction to trigonometry defining opposite, adjacent, and hypotenuse sides of a right triangle. 2. Exercises labeling the sides of example triangles and defining the trigonometric functions sine, cosine, and tangent. 3. More exercises calculating trigonometric functions based on given side lengths of triangles. 4. Examples solving for unknown side lengths using trigonometric functions. The document was written by Airil Bin Ahmad and covers basic trigonometry concepts and applications to right triangles.

1. Friday, January 25, 2013
TRIGONOMETRY
AIRIL BIN AHMAD

2. Friday, January 25, 2013
TRIGONOMETRY
BY
AIRIL BIN AHMAD
AIRIL BIN AHMAD

3. Friday, January 25, 2013
Trigonometry - Introduction
Opposite side Hypotenuse
Adjacent side
Adjacent side
Hypotenuse
Opposite side
AIRIL BIN AHMAD

4. Friday, January 25, 2013
Trigonometry - Introduction
Adjacent side
Opposite side
Hypotenuse
AIRIL BIN AHMAD

5. Friday, January 25, 2013
Trigonometry - Exercise
Name the side of the right angle
triangle given below.
1. B A 2. B A
C AB= Adjacent C AB= Opposite
BC= Opposite BC=Adjacent
AC= Hypotenuse AC=Hypotenuse
AIRIL BIN AHMAD

6. Friday, January 25, 2013
Trigonometry - Exercise
C
3. B A
4.
B A
C
AB= Opposite AB= Hypotenuse
BC= Hypotenuse BC= Adjacent
AC= Adjacent AC= Opposite
AIRIL BIN AHMAD

7. Friday, January 25, 2013
Trigonometry - Exercise
X
5. 6.
Z Y
Z
X
XY= Opposite Y
XY= Hypotenuse
YZ= Hypotenuse YZ= Opposite
XZ= Adjacent XZ= Adjacent
AIRIL BIN AHMAD

8. Friday, January 25, 2013
Trigonometry – Sin , Cos and Tan
Sin x0= Opposite
1. B Adjacent A
Hypotenuse
x0
Opposite Sin x0= BC
Hypotenuse
AC
C
AIRIL BIN AHMAD

9. Friday, January 25, 2013
Trigonometry – Sin , Cos and Tan
Cos x0= Adjacent
1. B Adjacent A
Hypotenuse
x0
Opposite Cos x0= AB
Hypotenuse
AC
C
AIRIL BIN AHMAD

10. Friday, January 25, 2013
Trigonometry – Sin , Cos and Tan
Tan x0= Opposite
1. B Adjacent A
Adjacent
x0
Opposite Tan x0= BC
Hypotenuse
AB
C
AIRIL BIN AHMAD

11. Friday, January 25, 2013
Trigonometry – Sin , Cos and Tan
Sin x0= Opposite
Hypotenuse
Cos x0= Adjacent
Hypotenuse
Tan x0= Opposite
Adjacent
AIRIL BIN AHMAD

12. Friday, January 25, 2013
Trigonometry – Sin , Cos and Tan
HANG LI PO
LEE PENG SAN
TONG SAM PAH
SOH CAH TOA
AIRIL BIN AHMAD

13. Friday, January 25, 2013
Trigonometry – Sin , Cos and Tan
S in Opposite H ypotenuse
Cos Adjacent H ypotenuse
Tan O pposite Adjacent
AIRIL BIN AHMAD

14. Friday, January 25, 2013
Trigonometry – Sin , Cos and Tan
3 4 5 8 15 17
5 12 13 9 12 15
6 8 10 10 24 26
7 24 25 12 16 20
AIRIL BIN AHMAD

15. Friday, January 25, 2013
Trigonometry - Exercise
Find the value of sin, cos and tan from the given triangle.
1. B A 2. B A
x0
x0
C Sin x0 = BC/AC AB/AC
C Sin x0 =
Cos x0 = AB/AC Cos x0 = BC/AC
Tan x0 = BC/AB Tan x0 = AB/BC
AIRIL BIN AHMAD

16. Friday, January 25, 2013
Trigonometry - Exercise
Find the value of sin, cos and tan from the given triangle.
3. Q 4. E
P
x0
x0
G
F
R
Sin x0 = P/R
Sin x0 = G/E
Cos x0 = Q/R Cos x0 = F/E
Tan x0 = P/Q Tan x0 = G/F
AIRIL BIN AHMAD

17. Friday, January 25, 2013
Trigonometry - Exercise
Find the value of sin, cos and tan from the given triangle.
5. 3 6
6.
x0
4 8
5 x0
Sin x0 = 4/5 Sin x0 = 6/10 = 3/5
Cos x0 = 3/5 Cos x0 = 8/10 = 4/5
Tan x0 = 4/3 Tan x0 = 6/8 = 3/4
AIRIL BIN AHMAD

18. Friday, January 25, 2013
TRIGONOMETRY
BY
AIRIL BIN AHMAD
AIRIL BIN AHMAD

19. Friday, January 25, 2013
Trigonometry - Exercise
2. Find the length of AB
1. Find the length of PR
and cos A
and sin y0
1. PR= 8cm
Sin y= 3/8
2. AB= 13cm
Cos A= 12/13
y0
AIRIL BIN AHMAD

20. Friday, January 25, 2013
Trigonometry - Exercise
3. Find sin y0
1. Sin y= 8/17 4. Find sin x0
and cos y0
2. Sin x= 7/ 25
17 cm
10cm
4cm
AIRIL BIN AHMAD

21. Friday, January 25, 2013
Trigonometry - Exercise
5. If cos x0 = 3/5. Find sin y0 6. Find sin y0
1. Sin y= 6/13
2. Sin y= 8/17
AIRIL BIN AHMAD

22. Friday, January 25, 2013
Trigonometry - Exercise
7. In the diagram, PQR and TSQ are straight 8. Given that sin x0= 8/10 and
lines. Given that tan x0 =6/9 and S is the length of PR = 21cm , find
the midpoint of TQ, find the length of PT and cos y0
sin y0
y0
AIRIL BIN AHMAD

23. Friday, January 25, 2013
Trigonometry - Exercise
9. In the figure, QS=8cm, find tan x0 10. In the figure, Find sin x0
S
T S
X0
R T
5
5
X0
Q P P
9 7 Q R
AIRIL BIN AHMAD

24. TRIGONOMETRY
BY
AIRIL BIN AHMAD
Friday, January 25, 2013 AIRIL BIN AHMAD

25. Friday, January 25, 2013
AIRIL BIN AHMAD

26. Friday, January 25, 2013
TRIGONOMETRY
Example 1
ABC is a right angle triangle. Given that tan y0 = 3 / 4, find the length of x and
sin y0
A
Sin y0 = Opposite
Hypotenuse
3 X= 15
x = 9 = 3
15 5
39
Y0
= C
B x 3 =
9 4 12
AIRIL BIN AHMAD

27. Friday, January 25, 2013
TRIGONOMETRY
Example 2
ABC is a right angle triangle. Given that tan y0 = 3 / 4, find the length of x and
sin y0
A
Sin y0 = Opposite
5 X 4 = 20 Hypotenuse
3 X= 20
x = 12 = 3
4 20 5
12
=
Y0
12 C
B x 4 =
4 16
AIRIL BIN AHMAD

28. Friday, January 25, 2013
TRIGONOMETRY
Exercise 1
ABC is a right angle triangle. Given that tan y0 = 5 / 12, find the length of x and
sin y0
A X = 26
Sin y0 = Opposite
Hypotenuse
X
= 10 = 5
26 13
Y0
C
B
24
AIRIL BIN AHMAD

29. Friday, January 25, 2013
TRIGONOMETRY
Exercise 2
ABC is a right angle triangle. Given that tan y0 = 4 / 3, find the length of x and
sin y0
A X = 40
Sin y0 = Opposite
Hypotenuse
X
= 18 = 3
30 5
Y0
C
B
24
AIRIL BIN AHMAD

30. Friday, January 25, 2013
TRIGONOMETRY
Exercise
1. In the diagram, PQRS is a rectangle. RST and QT are straight lines.
Given that tan x = 3 . Find sin y , cos y and tan y.
4
ANSWER:
x Sin y = 8/17
cos y = 15/17
Tan y = 8/15
AIRIL BIN AHMAD

31. Friday, January 25, 2013
TRIGONOMETRY
AIRIL BIN AHMAD

32. TRIGONOMETRY
EXERCISE
BY
AIRIL BIN AHMAD
Friday, January 25, 2013 AIRIL BIN AHMAD

33. Friday, January 25, 2013
TRIGONOMETRY
Exercise
1. In the diagram, PQRS is a rectangle. RST and QT are straight lines.
Given that tan x = 3 . Find sin y , cos y and tan y.
4
ANSWER:
x Sin y = 8/17
cos y = 15/17
Tan y = 8/15
AIRIL BIN AHMAD

34. Friday, January 25, 2013
QUESTION NO.1
In the diagram, S is the midpoint of PQ.
If tan x = 2 and PR = 10 cm, then tan y =
5
AIRIL BIN AHMAD

35. Friday, January 25, 2013
QUESTION NO.2
In the diagram, EFG and GHI are straight
lines.
If tan x = 2, then tan y =
AIRIL BIN AHMAD

36. Friday, January 25, 2013
QUESTION NO.3
3
In the diagram, sin x = . The value of y is
8
AIRIL BIN AHMAD

37. Friday, January 25, 2013
QUESTION NO.4
In the diagram, find the value of sin θ.
AIRIL BIN AHMAD

38. Friday, January 25, 2013
QUESTION NO.5
4
In the diagram, sin t = . Find the value of x.
5
AIRIL BIN AHMAD

39. Friday, January 25, 2013
QUESTION NO.6
In the diagram, the value of cos x is
24
25
AIRIL BIN AHMAD

40. Friday, January 25, 2013
QUESTION NO.7
In the diagram, LMN and PNQ are straight
3
lines. If cos x = ,
then find tan y. 5
AIRIL BIN AHMAD

41. Friday, January 25, 2013
QUESTION NO.8 5
In the diagram, tan x = , find value of cos y.
12
AIRIL BIN AHMAD

42. Friday, January 25, 2013
QUESTION NO.9
Given that tan 42° = 0.9, sin 42° = 0.67
and cos 42° = 0.74.
Calculate the height, in m,
of the tree shown in the diagram
AIRIL BIN AHMAD

43. Friday, January 25, 2013
QUESTION NO.10
The diagram shows a kite flying in the sky.
Given that tan 75° = 3.7, sin 75° = 0.97
and cos 75° = 0.26.
Find the value of t.
AIRIL BIN AHMAD

44. Friday, January 25, 2013
QUESTION NO.11
The diagram shows a flagpole that is
supported by a wire QR.
Find the angle of PRQ, in degree.
AIRIL BIN AHMAD

45. Friday, January 25, 2013
QUESTION NO.12
In the diagram, KLM is a straight line.
1
Given that tan x° = 4
1 and sin y° = 5
Calculate the length, 13
in cm, of LM.
AIRIL BIN AHMAD

46. Friday, January 25, 2013
QUESTION NO.13
In the diagram, GMH and MFN
are straight lines. If sin x° = 3
5
then find the value of tan y°.
AIRIL BIN AHMAD

47. Friday, January 25, 2013
QUESTION NO.14
The diagram shows a rectangle PQRS.
M is the midpoint of RS.
Find the value of cos x°.
AIRIL BIN AHMAD

48. Friday, January 25, 2013
QUESTION NO.15
In the diagram, JMK is a straight line.
3
Given that sin x° = 5 . Find the value of tan y°.
AIRIL BIN AHMAD

49. Friday, January 25, 2013
QUESTION NO.16
In the diagram, M is the
midpoint of QR.
4
Given that cos y° =
5
find the value of tan x°.
AIRIL BIN AHMAD

50. Friday, January 25, 2013
QUESTION NO.17
In the diagram, HJK and JML are straight lines.
Find the value of cos y°.
AIRIL BIN AHMAD

51. Friday, January 25, 2013
QUESTION NO.18
In the diagram,
find the value of
tan B° and cos B°.
AIRIL BIN AHMAD

52. Friday, January 25, 2013
QUESTION NO.19
In the diagram, FJMP is a rectangle. 6
JKS and PKM are straight lines. If tan x° =
5
then find the value of sin y°.
AIRIL BIN AHMAD

53. Friday, January 25, 2013
QUESTION NO.20
In the diagram, PMR is a straight line. M is the
midpoint of PR. Find the value of tan D0.
AIRIL BIN AHMAD

54. Friday, January 25, 2013
QUESTION NO.21
In the diagram, HJM and JPT are straight lines.
12
Given that cos x° = 13 , find the value of tan y o.
AIRIL BIN AHMAD

55. Friday, January 25, 2013
QUESTION NO.22
24
Based on the diagram, cos x° = , find FG.
25
AIRIL BIN AHMAD

56. Friday, January 25, 2013
QUESTION NO.23
In the diagram, PRMS is a straight line.
5
M is the midpoint of RS. Given that sin x° =
13
find the length of PR, in cm.
AIRIL BIN AHMAD

57. Friday, January 25, 2013
QUESTION NO.24
In the diagram, PST
and QTR are straight lines.
S is the midpoint of PT.
3
Given that tan yº =
4
find the length of PQ.
AIRIL BIN AHMAD

58. Friday, January 25, 2013
QUESTION NO.25
The diagram shows a square HJQS.
Find the value of sin x°.
AIRIL BIN AHMAD

59. Friday, January 25, 2013
QUESTION NO.26
In the diagram, PQR is a straight line.
3 , then find the value of tan yº.
If sin xº =
5
AIRIL BIN AHMAD

60. Friday, January 25, 2013
QUESTION NO.27
In the diagram, JPK is a straight line.
Given that JP = 7 cm,
calculate the value of tan xº.
AIRIL BIN AHMAD

61. Friday, January 25, 2013
QUESTION NO.28
In the diagram, PQR is a straight line.
3
Given that sin xº = . Find the length of PR.
5
30cm AIRIL BIN AHMAD

62. Friday, January 25, 2013
QUESTION NO.29
In Diagram 13, M is the midpoint of BC
in a right-angled triangle ABC. B
Given that cos x 7
= 25
Find the lengthof MC. m c
50 M
x
A C
AIRIL BIN AHMAD

63. Friday, January 25, 2013
QUESTION NO.30
In Diagram 9, a long pole is
placed against a wall.
x
The wall is 7 m high and
the foot of the pole is 24m
away from the foot of the wall.
Calculate the length, in m,
of pole that is jutting over
the wall (x), if the pole is
35 metres long. 7m
24 m
AIRIL BIN AHMAD

64. Friday, January 25, 2013
AIRIL BIN AHMAD

65. Friday, January 25, 2013
TRIGONOMETRY
(Using Calculator)
BY
AIRIL BIN AHMAD
AIRIL BIN AHMAD

66. Friday, January 25, 2013
TRIGONOMETRY
1 degree (10) = 60 min (600) 10=60’
Example 1
On your calculator
Change the angle in degrees
to degrees and minutes. 1. Press 30.8
a. 30.8 0 2. Press 0,,,
3. Press =
b. 15.30 = 150 18’
c. 28.50 = 280 30’ The value obtained is
300 48’
d. 50.90 = 500 54’
Friday, January 25, 2013 AIRIL BIN AHMAD

67. Friday, January 25, 2013
TRIGONOMETRY
Exercise Exercise
Change the angle in f. 79.90
degrees to degrees and g. 15.150
minutes. h. 30.650
a. 37.80 i. 47.250
b. 95.30 j. 9.550
c. 68.50 k. 36.450
d. 59.10 l. 80.350
e. 46.90
Friday, January 25, 2013 AIRIL BIN AHMAD

68. Friday, January 25, 2013
TRIGONOMETRY
Example 2 On your calculator
1. Press 30
Change the angle in degrees
and minutes to degrees 2. Press 0,,,
3. Press 18
a. 300 18’
4. Press 0,,,
b. 360 36’ = 36.6
5. Press =
c. 670 54’ = 67.9
6. Press 0,,,
d. 620 51’ = 62.85 The value obtained is
30.3
Friday, January 25, 2013 AIRIL BIN AHMAD

69. Friday, January 25, 2013
TRIGONOMETRY
Exercise Exercise
Change the angle in f. 790 23’
degrees and minutes to g. 150 42’
degrees. h. 300 51’
a. 300 27’ i. 470 17’
b. 150 54’ j. 90 9’
c. 280 24’ k. 360 24’
d. 500 12’ l. 800 33’
e. 360 08’
Friday, January 25, 2013 AIRIL BIN AHMAD

70. Friday, January 25, 2013
TRIGONOMETRY
Example 3
On your calculator
Use calculator to find the 1. Press sin
value of sin, cos and tan 2. Press 30
a. Sin 300 3. Press 0,,,
b. Sin 360 36’ = 0.60
4. Press =
c. Cos 670 = 0.39
d. Cos 620 51’ = 0.46
The value obtained is
e. Tan 350 = 0.70
0.5
f. Tan 740 20’ = 3.57
Friday, January 25, 2013 AIRIL BIN AHMAD

71. Friday, January 25, 2013
TRIGONOMETRY
Exercise Exercise
Find the value of: f. tan 790 23’
a. Sin 300 27’ g. tan 150 42’
b. tan 150 54’ h. sin 300 51’
c. cos 280 24’ i. cos 470 17’
d. cos 500 12’ j. Tan 90 9’
e. sin 360 08’ k. cos 360 24’
l. sin 800 33’
Friday, January 25, 2013 AIRIL BIN AHMAD

72. Friday, January 25, 2013
TRIGONOMETRY
Example 4 On your calculator
1. Press shift
Using calculator to find the value of 2. Press sin
the angle 3. Press 0.5925
a. Sin θ = 0.5925 4. Press =
b. Cos θ = 0.5925 = 530 39’
The value obtained is
c. Tan θ = 0.9342 = 43 03’
0
36.33460
d. Sin θ = 0.2479 = 140 21’
e. Sin θ = 0.7992 = 530 03’ 5. Press 0’’’
f. Cos θ = 0.3345 = 700 27’
The value is 360 20’
Friday, January 25, 2013 AIRIL BIN AHMAD

73. Friday, January 25, 2013
TRIGONOMETRY
Exercise Exercise
Find the value of: g. Tan θ = 0.1234
a. Sin θ = 0.8025 h. Sin θ = 0.4425
i. Cos θ = 0.5925
b. Cos θ = 0.2925
j. Tan θ = 2.9342
c. Tan θ = 1.8942 k. Sin θ = 0.5579
d. Sin θ = 0.0679 l. Sin θ = 0.0009
e. Sin θ = 0.9292 m. Cos θ = 0.7945
f. Cos θ = 0.0445
Friday, January 25, 2013 AIRIL BIN AHMAD

74. Friday, January 25, 2013
TRIGONOMETRY
Example 5 Solution
Find the angle of y0 Sin y0 = Opposite
A hypotenuse
sin y0 = 6 = 0.6
10cm
10
6cm
Y 0 y = 360 52’#
C
B
Friday, January 25, 2013 AIRIL BIN AHMAD

75. Friday, January 25, 2013
TRIGONOMETRY
Example 6 Solution
Find the angle of y0 Cos y0 = Adjacent
A hypotenuse
cos y0 = 9 = 0.6
15cm
15
Y 0 y = 530 07’#
C
B 9cm
Friday, January 25, 2013 AIRIL BIN AHMAD

76. Friday, January 25, 2013
TRIGONOMETRY
Exercise Exercise
1. Find the angle of y0 2. Find the angle of y0
A A
8cm 11cm
5cm 7cm
Y0 Y0
C C
B B
Friday, January 25, 2013 AIRIL BIN AHMAD

77. Friday, January 25, 2013
TRIGONOMETRY
Exercise Exercise
3. Find the angle of y0 4. Find the angle of y0
A A
9cm 7cm
Y0 Y0
C C
B 4cm B 3cm
Friday, January 25, 2013 AIRIL BIN AHMAD

78. Friday, January 25, 2013
TRIGONOMETRY
Exercise Exercise
5. Find the angle of y0 6. Find the angle of y0
A A
13cm 11cm
Y0 Y0
C C
B 6cm B 8cm
Friday, January 25, 2013 AIRIL BIN AHMAD

79. Friday, January 25, 2013
TRIGONOMETRY
Example 7 Solution
Find the length of x Sin 300 = Opposite
A hypotenuse
sin 300 = x
7cm
7
X 0.5 =x
300 7
B
C x = 0.5 x 7
x = 3.5cm
Friday, January 25, 2013 AIRIL BIN AHMAD

80. Friday, January 25, 2013
TRIGONOMETRY
Example 8 Solution
Find the length of x Sin 350 27’ = Opposite
A hypotenuse
sin 350 27’ = x
6cm
6
X 0.5800 =x
350 27’ 6
B
C x = 0.5800 x 6
x = 3.48cm
Friday, January 25, 2013 AIRIL BIN AHMAD

81. Friday, January 25, 2013
TRIGONOMETRY
Exercise Exercise
1. Find the length of x 2. Find the length of x
A A
X=10.39 X=12.82
12cm 15cm
x x
600 58045’
C C
B B
Friday, January 25, 2013 AIRIL BIN AHMAD

82. Friday, January 25, 2013
TRIGONOMETRY
Exercise Exercise
3. Find the length of x 4. Find the length of x
A A
X=6.22 X=6.23
7cm 13cm
27019’ 61023’
C C
B x B x
Friday, January 25, 2013 AIRIL BIN AHMAD

83. Friday, January 25, 2013
TRIGONOMETRY
Exercise Exercise
5. Find the length of x 6. Find the length of x
A A
X=17.15 X=9.02
x x
52050’ 48025’
C C
B 13cm B 8cm
Friday, January 25, 2013 AIRIL BIN AHMAD

84. Friday, January 25, 2013
QUESTION NO.9 7
Question
Given that tan 42° = 0.9, sin 42° = 0.67
and cos 42° = 0.74.
Calculate the height, in m,
of the tree shown in the diagram
Friday, January 25, 2013 AIRIL BIN AHMAD

85. Friday, January 25, 2013
QUESTION NO.10 8
Question
The diagram shows a kite flying in the sky.
Given that tan 75° = 3.7, sin 75° = 0.97
and cos 75° = 0.26.
Find the value of t.
Friday, January 25, 2013 AIRIL BIN AHMAD

86. Friday, January 25, 2013
QUESTION NO.11 9
Question
The diagram shows a flagpole that is
supported by a wire QR.
Find the angle of PRQ, in degree.
Friday, January 25, 2013 AIRIL BIN AHMAD

87. Friday, January 25, 2013
 10. A ladder makes an angle of 650 with a wall
when it reaches a height of 10m. up the wall.
How far is the foot of the ladder away from
the wall?
10m
650
 11. A straight road 240m long rises 120m
vertically from one end to the other. Find the
angle between the road and the horizontal.
120m
Friday, January 25, 2013 AIRIL BIN AHMAD
240