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11 Trigonometric Ratios 11.1 Introduction to Trigonometric Ratios

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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:


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