14 Trigonometric Ratios
14 Trigonometric Ratios
Activity
Activity 14.1 (p. 14.4)
4. The two ratios are the same and it is a constant.
Quick Practice
Quick Practice 14.1 (p. 14.5)
(a)
(b)
Quick Practice 14.2 (p. 14.6)
(a) (cor. to 3 sig. fig.)
(b) (cor. to 3 sig. fig.)
Quick Practice 14.3 (p. 14.7)
(a) (cor. to 3 sig. fig.)
(b) ∵
∴
Quick Practice 14.4 (p. 14.8)
(a)
(b)
(c)
Quick Practice 14.5 (p. 14.9)
∵
∴
Quick Practice 14.6 (p. 14.9)
∵
∴
Quick Practice 14.7 (p. 14.10)
∵
∴
Quick Practice 14.8 (p. 14.14)
(a)
(b)
Quick Practice 14.9 (p. 14.15)
(a) (cor. to 3 d.p.)
(b) (cor. to 3 d.p.)
(c) (cor. to 3 d.p.)
Quick Practice 14.10 (p. 14.16)
(a)
(b)
(c)
Quick Practice 14.11 (p. 14.17)
∵
∴
Quick Practice 14.12 (p. 14.18)
∵
∴
Quick Practice 14.13 (p. 14.18)
∵
∴
Quick Practice 14.14 (p. 14.22)
(a)
(b)
Quick Practice 14.15 (p. 14.23)
(a) (cor. to 4 sig. fig.)
(b) (cor. to 4 sig. fig.)
Quick Practice 14.16 (p. 14.24)
1.
2.
3.
Quick Practice 14.17 (p. 14.24)
∵
∴
Quick Practice 14.18 (p. 14.25)
∵
∴
Quick Practice 14.19 (p. 14.25)
∵
∴
∵
∴
Alternative solution
Quick Practice 14.20 (p. 14.29)
In △ABD,
In △ACD,
∴
Quick Practice 14.21 (p. 14.30)
Draw AE ⊥ CD as shown in the figure.
AE = BC = 8 cm
CE = BA = 6 cm
In △AED,
∴
Quick Practice 14.22 (p. 14.31)
Consider △OAC.
Area of the shaded region
Quick Practice 14.23 (p. 14.32)
In △ABE,
In △BCD,
∴ Length of the strip of metal
Quick Practice 14.24 (p. 14.37)
(a) Construct line segment OA such that OA makes an angle 10° with the positive x-axis.
(b) Construct line segment OB such that OB makes an angle 40° with the positive x-axis.
(c) Construct line segment OC such that OC makes an angle 55° with the positive x-axis.
(d) Construct line segment OD such that OD makes an angle 80° with the positive x-axis.
Further Practice
Further Practice (p. 14.10)
1. ∵
∴
2. ∵
∴
3. ∵
∴
4.
∴
Further Practice (p. 14.18)
1. ∵
∴
2. ∵
∴
3. ∵
∴
Further Practice (p. 14.26)
1. ∵
∴
2. ∵
∴
3. ∵
∴
Further Practice (p. 14.33)
1. In △ABC,
∴
In △ABD,
2. Draw AE ⊥ BC as shown in the figure.
AE = DC = 10 cm
EC = AD = 4 cm
In △ABE,
∴
∴
3. Draw AE ⊥ CD as shown in the figure.
ED = AB = 60 m
AE = BD = 20 m
In △AEC,
∴
Exercise
Exercise 14A (p. 14.11)
Level 1
1. (a)
(b)
(c)
(d)
2. (a)
(b)
3. (a)
(b) (Pyth. theorem)
4. (a) (cor. to 4 sig. fig.)
(b) (cor. to 4 sig. fig.)
(c) (cor. to 4 sig. fig.)
(d) (cor. to 4 sig. fig.)
(e) (cor. to 4 sig. fig.)
(f) (cor. to 4 sig. fig.)
5. (a)
(b)
(c)
(d)
(e)
(f)
6. (a)
(b)
(c)
7. (a)
(b)
(c)
8. (a)
(b)
(c)
9. (a)
(b)
(c)
Level 2
10. (a) (cor. to 4 sig. fig.)
(b) (cor. to 4 sig. fig.)
11. (a) (cor. to 4 sig. fig.)
(b) (cor. to 4 sig. fig.)
12. (a) (cor. to 4 sig. fig.)
(b) (cor. to 4 sig. fig.)
13. (a) (cor. to 4 sig. fig.)
(b) (cor. to 4 sig. fig.)
14. (a) (cor. to 4 sig. fig.)
(b) (cor. to 4 sig. fig.)
15. (a) (cor. to 3 sig. fig.)
(b) ∵
∴
16. (a)
(b)
(c)
(d)
17. (a)
(b)
18. (a)
(b)
19. (a) In △ABD,
In △ADC,
(b) In △QSR,
In △PQR,
20. (a) In △LMN,
In △MNK,
∴
(b) In △BCD,
In △ABD,
∴
Exercise 14B (p. 14.19)
Level 1
1. (a)
(b)
(c)
(d)
2. (a)
(b)
3. (a)
(b)
4. (a) (cor. to 4 sig. fig.)
(b) (cor. to 4 sig. fig.)
(c) (cor. to 4 sig. fig.)
(d) (cor. to 4 sig. fig.)
(e) (cor. to 4 sig. fig.)
(f) (cor. to 4 sig. fig.)
5. (a)
(b)
(c)
(d)
(e)
(f)
6. (a)
(b)
(c)
7. (a)
(b)
(c)
8. (a)
(b)
(c)
Level 2
9. (a) (cor. to 4 sig. fig.)
(b) (cor. to 4 sig. fig.)
(c) (cor. to 4 sig. fig.)
10. (a) (cor. to 4 sig. fig.)
(b) (cor. to 4 sig. fig.)
(c) (cor. to 4 sig. fig.)
11. (a) (cor. to 4 sig. fig.)
(b) (cor. to 4 sig. fig.)
(c) (cor. to 4 sig. fig.)
12. (a)
(b)
(c)
13. (a)
(b)
(c)
(d)
14. (a)
(b)
(c)
15. (a)
(b) ∵
∴
16. (a)
(b) ∵
∴
∵
∴
17. (a) In △BCD,
In △ABD,
(b) In △PRS,
In △PQS,
18. (a) In △MNK,
In △LMK,
∴
(b) In △ABD,
In △BCD,
∴
Exercise 14C (p. 14.26)
Level 1
1. (a)
(b)
(c)
2. (a)
(b)
(c)
3. (a) With the notations in the figure,
(b) With the notations in the figure,
(c) With the notations in the figure,
4. (a) (cor. to 4 sig. fig.)
(b) (cor. to 4 sig. fig.)
(c) (cor. to 4 sig. fig.)
5. (a)
(b)
(c)
(d)
(e)
(f)
6. (a)
(b)
(c)
7. (a)
(b)
(c)
8. (a)
(b)
(c)
Level 2
9. (a) (cor. to 4 sig. fig.)
(b) (cor. to 4 sig. fig.)
(c) (cor. to 4 sig. fig.)
10. (a) (cor. to 4 sig. fig.)
(b) (cor. to 4 sig. fig.)
(c) (cor. to 4 sig. fig.)
11. (a)
(b) (cor. to 4 sig. fig.)
(c) (cor. to 4 sig. fig.)
12. (a)
(b)
(c)
13. (a)
(b)
(c)
14. (a)
(b)
15. (a) In △ABD,
In △ADC,
(b) In △PSR,
In △PQR,
∴
16. (a) In △ABD,
In △BCD,
∴
(b) In △QRS,
In △PQR,
∴
17. (a) According to the question, AC is the longest side.
∵
∴ △ABC is a right-angled triangle, where ÐB is a right angle. (Converse of Pyth. theorem)
(b) ∵
∴
Exercise 14D (p. 14.33)
Level 1
1.
2.
3.
4. In △ABC,
∴
In △ABD,
5. In △PSQ,
∴
In △PRQ,
∴
6. In △ABC,
∴
In △ACD,
∴
7. In △PQR,
In △PRS,
∴
8. In △BAD,
∴
In △ABC,
∴
9. In △PQR,
In △PRS,
∴
10. (a) In △ABD,
(b) In △ABD,
In △ADC,
∴
11. In △PQR,
In △PRS,
∴ Area of quadrilateral PQRS
12. Draw EI ⊥ FG as shown in the figure.
∵ IG = EH = 7 cm
∴ FI = FG – IG = (14 – 7) cm = 7 cm
In △EFI,
∴ HG = EI = 7 tan 55° cm
∴ Perimeter of trapezium EFGH
13. (a) In △AOC,
∴
(b) Area of sector OAB
Level 2
14. (a)
In △OCD,
∴
(b) In △OCD,
Area of the shaded region
15.
16. (a)
(b) With the notations in the figure,
In △,
∴ The height of the balloon above the ground is
95 m.
17. BG = EF = 10 m
In △AGB,
In △CED,
∴ Total length of the slide
18. (a) Let l m be the distance between the horizontal ground and the top of the ladder.
∴ The distance between the horizontal ground and the top of the ladder is 4.70 m.
(b)
19. (a) In △BCD,
∴ ÐECF = ÐBCD = 40°
In △ECF,
(b) In △ABE,
(c) In △BCD,
∴
(d) In △ABE,
∵ ∠ECF = ∠EFC = 40°
∴ EF = EC (sides opp. equal Ðs)
∴
Revision Exercise 14 (p. 14.41)
Level 1
1. (a) (cor. to 3 sig. fig.)
(b) (cor. to 3 sig. fig.)
(c) (cor. to 3 sig. fig.)
(d) (cor. to 3 sig. fig.)
2. (a)
(b)
(c)
(d)
3. (a)
(b)
(c)
(d)
4.
5.
6.
7. ∵
∴
8. ∵
∴
9. ∵
∴
10. ∵
∴
11. ∵
∴
12. ∵
∴
13. ∵
∴
14. ∵
∴
15. ∵
∴
16. ∵
∴
17. ∵
∴
18. (a) In △PQR,
(b) In △PQR,
(c) In △PRS,
(d)
19. In △ABD,
In △BCD,
∴
20. In △LMN,
In △LNK,
∴
21. In △EFH,
∴
In △EFG,
∴
22. In △QRS,
∴
In △PQS,
∴
23. In △AOC,
∴ OB = OA = 5 cm
∴
Area of the shaded region
24. In △CFE,
In △DEF,
The length of the tunnel
Level 2
25. Draw AE ^ BC as shown in the figure.
In △ABE,
Area of parallelogram ABCD
26. Draw PT ^ QR as shown in the figure.
In △PQT,
Area of trapezium PQRS
27. Draw WQ and XR such that WQ⊥YZ and XR⊥WQ.
∵
∴ RX // QY (int. ∠s supp.)
In △WXR,
In △WQZ,
∴
28. (a) ∵ CFED is a square.
∴ DC = ED = x cm
∵ BD : DC = 5: 6
∴
(b) In △BDE,
∴
29. Draw DF ^ OA and join OD as shown in the figure.
In △COD,
∴ OF = 3 cm
In △DOF,
∴
Area of the shaded region
30. (a) In △BCD,
∴
∴ The angle that the pencil makes with BC is 59.0°.
(b) Draw EF⊥BF as shown in the figure.
In △BFE,
∴ The vertical distance from the top E of the pencil to the table is 13.7 cm.
31. (a) In △ADC,
∴ The distance between C and D is 12.9 m.
(b) In △ADC,
∴
In △BCD,
∴ The distance between B and C is 13.7 m.
32. In △ACK,
∴
In △BCK,
∴
∴
33. Draw BC⊥AP as shown in the figure.
In △ACB,
In △BCP,
∴ The distance between ship P and lighthouse B is 1891 m.
34. With the notations in the figure,
in △ACD,
∵ CB = AD = L cm
∴ CD + DB = CB
35. ∵
∴ ST // PR (int. ∠s supp.)
In △PQR,
Challenging Questions (p. 14.45)
1. Draw AF ^ BC as shown in the figure.
∵ ABC is an equilateral triangle.
∴ ∠ABC = 60° (prop. of equil. △)
In △ABF,
Area of △ABC
Area of △BCD
Difference between the areas of △BDE and △ACE
2. ∵ AD = CD
∴ ∠CAD =∠ACD (base ∠s, isos. △)
Let AB = x.
In △ABD,
∵ CD = AD = 2x
∴
In △ABC,
135