1. 11 Trigonometric Ratios
11.1 Introduction to Trigonometric Ratios
11.2 Sine Ratio
11.3 Cosine Ratio
11.4 Tangent Ratio
11.5 Trigonometric Ratios of Some Special Angles
11.6 Finding Trigonometric Ratios by Using Right-angled
Triangles
11.7 Basic Trigonometric Identities
11.8 Trigonometric Identities of Complementary Angles

2. 11.1 Introduction to Trigonometric Ratios

3. 11.1 Introduction to Trigonometric Ratios

4. 11.1 Introduction to Trigonometric Ratios

5. 11.2 Sine Ratio A. Concept of Sine Ratio
The reason for the triangles to be similar is AAA.

6. Example 1T Solution: 11 Trigonometric Ratios
In the figure C = 90, AB = 34, BC = 30 and AC = 16. Find sin A and sin B.
Solution:

7. B. Using a Calculator to Find sin q from q
11.2 Sine Ratio
B. Using a Calculator to Find sin q from q
The button is the same as the button in some other calculators.

8. Example 2T Solution: 11 Trigonometric Ratios
Using a calculator, find the values of the following expressions correct to 3 significant figures.
(a) 1 – sin 56 (b) sin 23.4 + 0.5
Solution:
(a) Key-in Sequence: 1 –
sin 56
EXE
Display: ∴
(b) Key-in Sequence:
sin
23.4 +
0.5
EXE
Display: ∴

9. Example 3T Solution: 11 Trigonometric Ratios
(a) Use a calculator to find the value of (sin 72 – sin 12) – sin 60,
correct to 3 significant figures.
(b) Is sin 72 – sin 12 equal to sin 60? Explain your answer.
Solution:
(a) (sin 72 – sin 12) – sin 60
(b) ∵ (sin 72 – sin 12) – sin 60  0
∴ sin 72 – sin 12  sin 60

10. 11.2 Sine Ratio
B. Using a Calculator to Find sin q from q

11. 11.2 Sine Ratio
C. Using a Calculator to Find q from sin q

12. Example 4T Solution: 11 Trigonometric Ratios
In the following expressions, find q correct to 3 significant figures.
(a) sin q  sin 16  sin 46 (b)
Solution:
(a) Key-in Sequence: 16 46
Display:
sin +
sin
EXE
SHIFT
sin
EXE
Display:
∴ q  84.3 (cor. to 3 sig. fig.)
(b) Key-in Sequence:
sin 33 
sin 77
EXE
Display:
Display:
SHIFT
sin
EXE
∴ q  34.0 (cor. to 3 sig. fig.)

13. 11.2 Sine Ratio
D. Using Sine Ratios to Find Unknowns in
Right-angled Triangles

14. Example 5T Solution: 11 Trigonometric Ratios
In the figure, A  18, B  90 and AC  10 cm. Find BC correct to 2 decimal places.
Solution:
sin A
sin 18
BC  10 sin 18 cm
 3.09 cm (cor. to 2 d. p.)

15. Example 6T Solution: 11 Trigonometric Ratios
In the figure, B  90, C  10 and AB  5 cm. Find AC correct to 1 decimal place.
Solution:
sin C
sin 10 AC
 28.8 cm (cor. to 1 d. p.)

16. Example 7T Solution: 11 Trigonometric Ratios
In the figure, B  90, AB  6 cm and AC  10 cm. Find C correct to 3 significant figures.
Solution:
sin C
C  36.9 (cor. to 3 sig. fig.)

17. 11.3 Cosine Ratio
A. Concept of Cosine Ratio

18. Example 8T Solution: 11 Trigonometric Ratios
In the figure, B  90, AB  10, BC  24 and AC  26. Find cos A and cos C.
Solution:
cos A
cos C

19. B. Using a Calculator to Find cos q from q
11.3 Cosine Ratio
B. Using a Calculator to Find cos q from q
You will learn more about the importance of sin2q and cos2q in Section 11.7.

20. Example 9T Solution: 11 Trigonometric Ratios
Using a calculator, find the value of 7 cos 25.6  3 cos2 70 correct to 3 significant figures.
Solution:
Key-in Sequence: 7
25.6 3 70
cos + (
cos ) x2
EXE ∴

21. Example 10T Solution: 11 Trigonometric Ratios
(a) Use a calculator to find the value of (cos 44 – cos 16) – cos 28,
correct to 3 significant figures.
(b) Is cos 44 – cos 16 equal to cos 28? Explain your answer.
Solution:
(a) (cos 44 – cos 16) – cos 28
(b) ∵ (cos 44 – cos 16) – cos 28  0
∴ cos 44 – cos 16  cos 28

22. 11.3 Cosine Ratio
B. Using a Calculator to Find cos q from q

23. 11.3 Cosine Ratio
C. Using a Calculator to Find q from cos q

24. Example 11T Solution: 11 Trigonometric Ratios
In the following expressions, find q correct to 3 significant figures.
(a) cos q  cos 17 – cos 37 (b)
Solution:
(a) Key-in Sequence: 17 37
Display:
cos –
cos
EXE
SHIFT
cos
EXE
Display:
∴ q  80.9 (cor. to 3 sig. fig.)
(b) Key-in Sequence:
cos 49 
cos 13
EXE
Display:
Display:
SHIFT
cos
EXE
∴ q  47.7 (cor. to 3 sig. fig.)

25. 11.3 Cosine Ratio
D. Using Cosine Ratios to Find Unknowns in
Right-angled Triangles

26. Example 12T Solution: 11 Trigonometric Ratios
In the figure, B  90, C  26 and AC  20 cm. Find BC correct to 2 decimal places.
Solution:
cos C
cos 26
BC  20 cos 26 cm
 cm (cor. to 2 d. p.)

27. Example 13T Solution: 11 Trigonometric Ratios
In the figure, A  18, C  90 and AC  9 cm. Find AB correct to 1 decimal place.
Solution:
cos A
 9.5 cm (cor. to 1 d. p.)

28. Example 14T Solution: 11 Trigonometric Ratios
In the figure, C  90, AB  17 cm and AC  11 cm. Find A correct to 3 significant figures.
Solution:
(cor. to 3 sig. fig.)

29. 11.4 Tangent Ratio
A. Concept of Tangent Ratio

30. Example 15T Solution: 11 Trigonometric Ratios
In the figure, A  90, AB  7, BC  25 and AC  24. Find tan B and tan C.
Solution:
tan B
tan C

31. 11.4 Tangent Ratio
A. Concept of Tangent Ratio

32. 11.4 Tangent Ratio
B. Using a Calculator to Find tan q from q

33. Example 16T Solution: 11 Trigonometric Ratios
Using a calculator, find the value of 3 tan 31.8 tan 45.5 correct to 3 significant figures.
Solution:
Key-in Sequence: 3
31.8
45.5
Display:
tan
tan
EXE ∴

34. 11.4 Tangent Ratio
B. Using a Calculator to Find tan q from q

35. 11.4 Tangent Ratio
C. Using a Calculator to Find q from tan q

36. Example 17T Solution: 11 Trigonometric Ratios
In the following expressions, find q correct to 3 significant figures.
(a) tan q  tan 70.5 + tan 35 (b) tan q  tan 62.1 tan 84
Solution:
(a) Key-in Sequence:
70.5 35
Display:
tan +
tan
EXE
SHIFT
sin
EXE
Display:
∴ q  74.2 (cor. to 3 sig. fig.)
(b) Key-in Sequence:
tan
62.1
tan 84
EXE
Display:
Display:
SHIFT
sin
EXE
∴ q  86.8 (cor. to 3 sig. fig.)

37. 11.4 Tangent Ratio
D. Using Tangent Ratios to Find Unknowns in
Right-angled Triangles

38. Example 18T Solution: 11 Trigonometric Ratios
In the figure, A  90, B  52 and AB  11 cm. Find AC correct to 3 significant figures.
Solution:
tan B
tan 52
AC  11 tan 52 cm
 14.1 cm (cor. to 3 sig. fig.)

39. Example 19T Solution: 11 Trigonometric Ratios
In the figure, A  48, B  90 and BC  25 cm. Find AB correct to 1 decimal place.
Solution:
tan A
tan 48 AB
 22.5 cm (cor. to 1 d. p.)

40. Example 20T Solution: 11 Trigonometric Ratios
In the figure, C  90, BC  14 cm and AC  13 cm. Find A correct to 3 significant figures.
Solution:
tan A
tan 48 A
 47.1 (cor. to 3 sig. fig.)

41. 11.5 Trigonometric Ratios of Some Special Angles

42. 11.5 Trigonometric Ratios of Some Special Angles

43. Example 21T Solution: 11 Trigonometric Ratios
In the figure, ABCD is a square with sides of cm. BC is produced to E such that E  60. Find the perimeter of ABED.
(Leave the answer in surd form.)
Solution:
∵ ABCD is a square.
∴ CD  BC  AB = cm
In CDE,
∴ Perimeter of ABED

44. Example 22T Solution: 11 Trigonometric Ratios
Without using a calculator, find the values of the following expressions. Leave the answers in surd form if necessary.
(a) (b)
Solution:
(a)
(b)

45. 11.5 Trigonometric Ratios of Some Special Angles

46. Example 23T Solution: 11 Trigonometric Ratios
Without using a calculator, solve the following trigonometric equations.
(a) 4 tan q  3 = (b)
Solution:
(a) 4tan  3  7
(b)
4tan  4
tan  1

47. 11.6 Finding Trigonometric Ratios by Using Right-angled Triangles

48. Example 24T Solution: AB 11 Trigonometric Ratios
Given that tan q  0.75, find the values of sin q and cos q without finding q . (Give the answers in fraction form.)
Solution:
Rewrite the given ratio in fraction form
Construct DABC as shown in the figure. AB
(Pyth. theorem) 

49. Example 25T Solution: AC   11 Trigonometric Ratios
Given that cos q  , find the values of sin q tan2 q without finding q .
Solution:
Construct DABC as shown in the figure. AC
(Pyth. theorem)  

50. 11.7 Basic Trigonometric Identities
Alternate forms of the second trigonometric identity:
(a) sin2 q  1 – cos2 q
(b) cos2 q  1 – sin2 q

51. 11.7 Basic Trigonometric Identities

52. Example 26T Solution: 11 Trigonometric Ratios
Simplify the following expressions.
(a) (b) cos q sin q tan q – 1
Solution:
(b) cos q sin q tan q – 1
(a)

53. 11 Trigonometric Ratios
Example 27T
Simplify
Solution:

54. Example 28T Solution: 11 Trigonometric Ratios
(a) Rewrite in terms of only.
(b) Given that , find the value of 3tan2 q without finding q.
Solution:
(a)
(b)

55. Example 29T Solution: 11 Trigonometric Ratios
Given that tan q = 0.4, find the value of without finding q.
Solution:
From the identity , we have ∴

56. Example 30T Solution: 11 Trigonometric Ratios
Without using a calculator, solve 4 cos q – 3 sin q  sin q .
Solution:
4 cos q – 3 sin q  sin q
4 cos q  4 sin q
tan q  1
q  45

57. 11 Trigonometric Ratios
Example 31T
Prove that
Solution:
L.H.S. ∴

58. 11.8 Trigonometric Identities of Complementary Angles

59. 11 Trigonometric Ratios
Example 32T
Simplify
Solution:

60. 11.8 Trigonometric Identities of Complementary Angles
The alternate form of the trigonometric identities on the previous page:
1. sin q  cos (90 – q)
2. cos q  sin (90 – q)
3. tan q 

61. Example 33T Solution: 11 Trigonometric Ratios
Without using a calculator, find the values of the following expressions.
(a) sin2 22  sin2 68 (b)
Solution:
(a) sin2 22  sin2 68  sin2 22  cos2 (90 – 68)
 sin2 22  cos2 22
 1
(b)
 tan (90 – 13) – tan 77
 tan 77 – tan 77
 0

62. Example 34T Solution: 11 Trigonometric Ratios
Without using a calculator, solve 2 tan  tan 19 = 2.
Solution:

63. Example 35T Solution:  11 Trigonometric Ratios
Prove that sin  cos  tan (90 – )  sin2 (90 – ).
Solution: 

64. Follow-up 1 Solution: 11 Trigonometric Ratios
In each of the following figures, find sin A and sin C.
(a) (b)
Solution:
(b) sin A
(a) sin A
sin C
sin C

65. Follow-up 2 Solution: 11 Trigonometric Ratios
Using a calculator, find the values of the following expressions correct to 3 significant figures.
(a) (b) 8 sin 12.5
Solution:
(a) Key-in Sequence:
1.6 35
Display: 
sin
EXE ∴
(b) Key-in Sequence: 8
12.5
Display:
sin
EXE ∴
8 sin 12.5

66. Follow-up 3 Solution: 11 Trigonometric Ratios
(a) Use a calculator to find the value of (sin 20 + sin 30) – sin 50,
correct to 3 significant figures.
(b) Is sin 20 + sin 30 equal to sin 50? Explain your answer.
Solution:
(a) (sin 20 + sin 30) – sin 50
(b) ∵ (sin 20 + sin 30) – sin 50  0
∴ sin 20 + sin 30  sin 50

67. Follow-up 4 Solution: 11 Trigonometric Ratios
In the following expressions, find q correct to 3 significant figures.
(a) sin q  2 sin 22.2 (b)
Solution:
(a) Key-in Sequence: 2
22.2
Display:
sin
EXE
SHIFT
sin
EXE
Display:
∴ q  49.1 (cor. to 3 sig. fig.)
(b) Key-in Sequence:
sin 15 
sin 40
EXE
Display:
Display:
SHIFT
sin
EXE
∴ q  23.7 (cor. to 3 sig. fig.)

68. Follow-up 5 Solution: 11 Trigonometric Ratios
In the figure, B  74, C  90 and AB  14 cm. Find AC correct to 1 decimal place.
Solution:
sin B
sin 74
AC  14 sin 74 cm
 13.5 cm (cor. to 1 d. p.)

69. Follow-up 6 Solution: 11 Trigonometric Ratios
In the figure, A  90, C  67 and AB  8 cm. Find BC correct to 1 decimal place.
Solution:
sin C
sin 67 BC
 8.7 cm (cor. to 1 d. p.)

70. Follow-up 7 Solution: 11 Trigonometric Ratios
In the figure, B  90, BC  9 cm and AC  12 cm. Find A correct to 3 significant figures.
Solution:
sin A
A  48.6 (cor. to 3 sig. fig.)

71. Follow-up 8 Solution: 11 Trigonometric Ratios
In each of the following figures, find cos B and cos C.
(a) (b)
Solution:
(a) cos B
cos C
(b) cos B
cos C

72. Follow-up 9 Solution: 11 Trigonometric Ratios
Using a calculator, find the value of the following expressions correct to 3 significant figures.
(a) cos 8.6  3 cos2 39.4 (b) 6 cos2 40 – 2 sin 70
Solution:
(a) Key-in Sequence:
8.6 3
39.4
cos  (
cos )
x 2
EXE ∴
(b) Key-in Sequence: 6 40 2 70 (
cos )
x 2 
sin
EXE ∴

73. Follow-up 10 Solution: 11 Trigonometric Ratios
(a) Use a calculator to find the value of (cos 70 + cos 10) – cos 80,
correct to 3 significant figures.
(b) Is cos 70 + cos 10 equal to cos 80? Explain your answer.
Solution:
(a) (cos 70 + cos 10) – cos 80
(b) ∵ (cos 70 + cos 10) – cos 80  0
∴ cos70  cos10  cos80

74. Follow-up 11 Solution: 11 Trigonometric Ratios
In the following expressions, find q correct to 3 significant figures.
(a) cos q  1 – 3 cos 76.9 (b)
Solution:
(a) Key-in Sequence: 1 3
76.9
Display: –
cos
EXE
SHIFT
cos
EXE
Display:
∴ q  71.3 (cor. to 3 sig. fig.)
(b) Key-in Sequence:
cos 66 
cos 33
EXE
Display:
Display:
SHIFT
cos
EXE
∴ q  61.0 (cor. to 3 sig. fig.)

75. Follow-up 12 Solution: 11 Trigonometric Ratios
In the figure, A  56, B  90 and AC  8 cm.
Find AB correct to 2 decimal places.
Solution:
cos A
cos 56
AB  8 cos 56 cm
 4.47 cm (cor. to 2 d. p.)

76. Follow-up 13 Solution: 11 Trigonometric Ratios
In the figure, A  40, B  90 and AB  16 cm.
Find AC correct to 1 decimal place.
Solution:
cos A
cos 40 AC
 20.9 cm (cor. to 1 d. p.)

77. Follow-up 14 Solution: 11 Trigonometric Ratios
In the figure, B  90, BC  15 cm and AC  60 cm.
Find C correct to 3 significant figures.
Solution:
cos C
C  75.5 (cor. to 3 sig. fig.)

78. Follow-up 15 Solution: 11 Trigonometric Ratios
In each of the following figures, find tan A and tan B.
(a) (b)
Solution:
(a) tan A
tan B
(b) tan A
tan B

79. Follow-up 16 Solution: 11 Trigonometric Ratios
Using a calculator, find the values of the following expressions correct to 3 significant figures.
(a) tan 5.78 – 2 tan 67 (b)
Solution:
(a) Key-in Sequence:
5.78 2 67
tan 
tan
EXE ∴
(b) Key-in Sequence: 46 1
3.14
tan 
EXE 
tan
EXE ∴

80. Follow-up 17 Solution: 11 Trigonometric Ratios
In the following expressions, find q correct to 3 significant figures.
(a) tan q  (b) tan q  tan 53 – tan 29
Solution:
(a) Key-in Sequence: 12 38
Display:
tan +
tan
EXE
SHIFT
tan (
Ans  6 )
EXE
∴ q  9.41 (cor. to 3 sig. fig.)
(b) Key-in Sequence:
tan 53 –
tan 29
EXE
Display:
Display:
SHIFT
tan
EXE
∴ q  37.7 (cor. to 3 sig. fig.)

81. Follow-up 18 Solution: 11 Trigonometric Ratios
In the figure, B  90, C  70 and BC  3 cm. Find AB correct to 3 significant figures.
Solution:
tan C
tan 70
AB  3 tan 70 cm
 8.24 cm (cor. to 3 sig. fig.)

82. Follow-up 19 Solution: 11 Trigonometric Ratios
In the figure, A  90, B  53 and AC  17 cm. Find AB correct to 3 significant figures.
Solution:
tan B
tan 53 AB
 12.8 cm (cor. to 3 sig. fig)

83. Follow-up 20 Solution: 11 Trigonometric Ratios
In the figure, B  90, AB  12 cm and BC  16 cm. Find A correct to 3 significant figures.
Solution:
tan A
A  53.1 (cor. to 3 sig. fig.)

84. Follow-up 21 Solution: 11 Trigonometric Ratios
In the figure, D is a point on AC such that BD  AC. A  45, CBD  60 and AB  cm. Find AC.
(Leave the answer in surd form.)
Solution:
In DABD,
In DBCD,
∴ AC

85. Follow-up 22 Solution: 11 Trigonometric Ratios
Without using a calculator, find the values of the following expressions. Leave the answers in surd form if necessary.
(a) (b) tan45(sin45  cos45)
Solution:
(a)
(b) tan45(sin45  cos45)

86. Follow-up 23 Solution: 11 Trigonometric Ratios
Without using a calculator, solve the following trigonometric equations.
(a) (b) 2 sin q = tan 60
Solution:
(a)
(b) 2 sin q = tan 60

87. Follow-up 24 Solution: 11 Trigonometric Ratios
Given that tan q  2.4, find the values of sin q and cos q without finding q . (Give the answers in fraction form.)
Solution:
Rewrite the given ratio in fraction form:
Construct DABC as shown in the figure. AB
(Pyth. theorem) ∴

88. Follow-up 25 Solution: 11 Trigonometric Ratios
Given that tan q  , find the value of sin q cos q without finding q .
Solution:
Construct DABC as shown in the figure. AB ∴ ∴

89. Follow-up 26 Solution: 11 Trigonometric Ratios
Simplify the following expressions.
(a) (b)
Solution:
(a)

90. Follow-up 26 Solution: 11 Trigonometric Ratios
Simplify the following expressions.
(a) (b)
Solution:
(b)

91. 11 Trigonometric Ratios
Follow-up 27
Simplify
Solution:

92. Follow-up 28 Solution: 11 Trigonometric Ratios
(a) Rewrite in terms of cos2 q only.
(b) Given that cos q = 0.75, find the value of sin2 q - 3 cos2 q without finding q.
Solution:
(a)
(b)

93. Follow-up 29 Solution: 11 Trigonometric Ratios
Given that tan q  , find the value of without finding q.
Solution:
From the identity , we have ∴

94. Follow-up 30 Solution: 11 Trigonometric Ratios
Without using a calculator, solve
Solution:

95. Follow-up 31 Solution: 11 Trigonometric Ratios
Prove the following trigonometric identities.
(a) (b)
Solution:
(a) ∴

96. Follow-up 31 Solution: 11 Trigonometric Ratios
Prove the following trigonometric identities.
(a) (b)
Solution:
(b) ∴

97. 11 Trigonometric Ratios
Follow-up 32
Simplify
Solution:

98. Follow-up 33 Solution: 11 Trigonometric Ratios
Without using a calculator, find the values of the following expressions.
(a) (b)
Solution:
(a)
(b)

99. Follow-up 34 Solution: 11 Trigonometric Ratios
Without using a calculator, solve
Solution:

100. 11 Trigonometric Ratios
Follow-up 35
Prove that
Solution: ∴