第1章：有向數



Chapter 7 Trigonometry

|Warm-up Exercise |

1. Simplify the following.

(a) [pic] (b) [pic]

2. Rationalize the following.

(a) [pic] (b) [pic]

3. Which of the following is a right-angled triangle? Explain your answer.

(a) [pic] (b) [pic]

4. Find the values of x in the following figures.

(a) [pic] (b) [pic]

5. Simplify the following.

(a) [pic] (b) [pic]

6. Simplify the following.

(a) [pic] (b) [pic]

|Build-up Exercise |

[ This part provides three extra sets of questions for each exercise in the textbook, namely Elementary Set, Intermediate Set and Advanced Set. You may choose to complete any ONE set according to your need. ]

Exercise 7A

((((((((((((((( Elementary Set [pic] (((((((((((((((

Level 1

1. In each of the following figures, find (. (Correct your answers to 1 decimal place.)

(a) [pic] (b) [pic]

(c) [pic]

In each of the following, find sin (, cos ( and tan (. Leave your answers in surd form if necessary. (2 ( 5)

2. [pic] 3. [pic]

4. [pic] 5. [pic]

6. Find the value of sin 30( ( cos 60(.

7. If [pic] where 0( ( ( ( 90(, find sin ( and tan (.

8. If [pic] where 0( ( ( ( 90(, find cos ( and tan (.

9. If tan ( ( 2 where 0( ( ( ( 90(, find sin ( and cos (.

Level 2

10. If [pic] where 0( ( ( ( 90(, find sin ( ( tan (.

11. Find ( such that [pic] where 0( ( ( ( 90(.

12. Find ( such that [pic] where 0( ( ( ( 90(. (Correct your answer to 1 decimal place.)

13. In the figure, (ABC is a right-angled triangle. AB ( 9 cm, AC ( 6 cm, BC ( CD and (ABD ( 85(.

(a) Find (ABC.

(b) Find (CBD.

(c) Find the length of CD.

(Correct your answers to 1 decimal place.)

((((((((((((((( Intermediate Set [pic] (((((((((((((((

Level 1

14. In the figure, find (. (Correct your answer to 1 decimal place.)

[pic]

In each of the following, find sin (, cos ( and tan (. Leave your answers in surd form if necessary. (15 ( 18)

15. [pic] 16. [pic]

17. [pic] 18. [pic]

19. In the figure, find the unknowns. (Correct your answers to 1 decimal place.)

[pic]

20. If [pic] where 0( ( ( ( 90(, find cos ( and tan (.

21. Find the value of [pic]. (Leave your answer in surd form.)

Level 2

22. If [pic] where 0( ( ( ( 90(, find [pic].

23. If [pic] where 0( ( ( ( 90(, find sin2( ( 2 cos2(.

24. Find ( such that [pic] where 0( ( ( ( 90(.

25. Find ( such that [pic] where 0( ( ( ( 90(. (Correct your answer to 1 decimal place.)

26. In the figure, AB ( 8 cm, AD ( 5 cm and

(ABC ( (BAD ( (BDC ( 90(.

(a) Find (DBC.

(b) Find the length of DC.

(Correct your answers to 1 decimal place.)

27. In the figure, ABCDE is a straight line and CF ( AE.

(a) Find the length of AE.

(b) Find (AFE.

(Correct your answers to 1 decimal place.)

(((((((((((((((( Advanced Set [pic] ((((((((((((((((

Level 1

In each of the following, find sin (, cos ( and tan (. Leave your answers in surd form if necessary. (28 ( 31)

28. [pic] 29. [pic]

30. [pic] 31. BCD is a straight line.

32. In the figure, BCD is a straight line. Find the unknowns. (Correct your answers to 1 decimal place.)

33. Find the value of [pic]. (Leave your answer in surd form.)

Level 2

34. If [pic] where 0( ( ( ( 90(, find 2 sin ( cos (. (Leave your answer in surd form.)

35. If tan( ( 3 where 0( ( ( ( 90(, find [pic].

36. Find ( such that sin 3( ( 1 where 0( ( ( ( 90(.

37. Find ( such that [pic] where 0( ( ( ( 90(.

38. In the figure, (ACF and (CEF are two right-angled triangles. AB ( 150 m, DE ( 200 m, CF ( 180 m, (FAB ( 30( and (FED ( 20(. Find the value of x. (Correct your answer to the nearest integer.)

39. In the figure, AFE and BCDE are straight lines where CD ( 15 cm and DE ( 25 cm. If AC ( BE, DF ( AE, AB ( 20 cm and (ABC ( 50(,

(a) find the length of AD.

(b) find (DAF.

(c) find the length of DF.

(Correct your answers to 1 decimal place.)

40. In the figure, A and B are two points in a valley lying 200 m apart and are opposite to each other. Their altitudes above the bottom C of the valley are both h m. If (BAC ( 50( and (ABC ( 40(, find

(a) the distance between C and B.

(b) the distance between A and C.

(c) the value of h.

(Correct your answers to 1 decimal place.)

41. In the figure, two ladders AB and AC lean against the steps of a stair. The height and width of each step are 15 cm and 25 cm respectively. If (DAB ( 30(,

(a) find the length of AB.

(b) find (DAC.

(Correct your answers to 1 decimal place if necessary.)

Exercise 7B

[ In this exercise, leave your answers in surd form if necessary. ]

((((((((((((((( Elementary Set [pic] (((((((((((((((

Level 1

1. In each of the following figures, find (.

(a) [pic] (b) [pic]

(c) [pic]

2. Find the reference angle corresponding to each of the following angles.

(a) 123( (b) 217( (c) 297(

3. If sin ( ( 0 and tan ( ( 0, determine the quadrant in which ( lies.

4. Find sin (, cos ( and tan ( in the figure.

[pic]

5. Express sin 160( in terms of acute angle.

6. Express cos(20( + 150() in terms of acute angle.

Find the value of each of the following trigonometric ratios. (7 ( 9)

7. tan 135( 8. sin 210( 9. cos ((225()

Level 2

Find the value of each of the following expressions. (10 ( 12)

10. sin 45(cos120( 11. 2 sin 135( ( 3 cos 225( 12. 2 (cos 330( ( sin 150()

((((((((((((((( Intermediate Set [pic] (((((((((((((((

Level 1

13. Find the reference angle corresponding to each of the following angles.

(a) 112( (b) 180( (c) 329(

14. If sin ( cos ( ( 0, determine the quadrant(s) in which ( lies.

15. Find sin (, cos ( and tan ( in the figure.

[pic]

16. Express tan 350( in terms of acute angle.

17. Express sin(250( ( 30() in terms of acute angle.

Find the value of each of the following trigonometric ratios. (18 ( 20)

18. sin 225( 19. cos 330( 20. tan ((135()

Level 2

Find the value of each of the following expressions. (21 ( 25)

21. tan 330( sin 300( 22. 5 tan 45( ( 6 tan 315(

23. [pic] 24. [pic]

25. cos 120( sin 225( ( sin 330( cos 225(

(((((((((((((((( Advanced Set [pic] ((((((((((((((((

Level 1

26. If [pic], determine the quadrant(s) in which ( lies.

27. Find sin (, cos ( and tan ( in the figure.

[pic]

28. Express cos ((120() in terms of acute angle.

29. Express tan ((130( ( 210() in terms of acute angle.

Find the value of each of the following trigonometric ratios. (30 ( 32)

30. tan ((30() 31. tan 405( 32. (cos ((210()

Level 2

Find the value of each of the following expressions. (33 ( 36)

33. sin ((30() cos ((120() 34. 3 cos ((60() ( 4 tan ((225()

35. [pic] 36. [pic]

37. In the figure, AOC is a straight line. ADB is a straight line perpendicular to the x-axis and AB ( BC. Find sin (.

[pic]

Exercise 7C

[ In this exercise, leave your answers in surd form if necessary. ]

((((((((((((((( Elementary Set [pic] (((((((((((((((

Level 1

1. If sin ( ( 0, find the range of ( where 0( ( ( ( 360(.

2. If cos ( ( 0, find the range of ( where 0( ( ( ( 360(.

3. If tan ( ( 0, find the range of ( where 0( ( ( ( 360(.

4. Find sin ( and tan ( if [pic] where 90( ( ( ( 180(.

5. Find cos ( and tan ( if [pic] where 270( ( ( ( 360(.

6. Find sin ( and cos ( if [pic] where 180( ( ( ( 270(.

7. Find sin ( and cos ( if [pic] where ( lies in quadrant III.

8. Find sin( and tan( if [pic] where ( lies in quadrant II.

9. Find cos ( and tan ( where ( lies in quadrant IV if [pic].

Level 2

10. If [pic] and 180( ( ( ( 270(, find the value of 3 sin ( ( 2 cos (.

11. If [pic] and 90( ( ( ( 180(, find the value of sin ( tan (.

12. If [pic] and 270( ( ( ( 360(, find the value of 4 tan ( ( sin (.

13. If tan ( ( (1 and cos ( ( 0, find sin (.

((((((((((((((( Intermediate Set [pic] (((((((((((((((

Level 1

14. Find sin ( and tan ( if [pic] where 270( ( ( ( 360(.

15. Find cos ( and tan ( if [pic] where 180( ( ( ( 270(.

16. Find sin ( and cos ( if [pic] where 90( ( ( ( 180(.

17. Find sin ( and cos ( if [pic] where ( lies in quadrant IV.

18. Find sin ( and tan ( if [pic] where ( lies in quadrant III.

19. Find cos ( and tan ( if [pic] where ( lies in quadrant II.

Level 2

20. If [pic] and 270( ( ( ( 360(, find the value of [pic].

21. If [pic] and 90( ( ( ( 180(, find the value of [pic].

22. If [pic] and 180( ( ( ( 270(, find the value of [pic].

23. If [pic] and tan ( ( 0, find the value of [pic].

24. (a) If cos ( ( k and 270( ( ( ( 360(, express sin ( and tan ( in terms of k.

(b) If sin ( ( k and 90( ( ( ( 180(, express cos ( and tan ( in terms of k.

(c) If [pic] and 180( ( ( ( 270(, express sin ( and cos ( in terms of k.

25. If [pic], [pic] and both A and B are obtuse angles, find the values of the following.

(a) [pic] (b) [pic]

(c) [pic]

26. A(1, 0) and B(x, y) are two points on the unit circle with centre at the origin and (AOB ( (. If y ( 2x, find the possible values of sin (, cos ( and tan (.

(((((((((((((((( Advanced Set [pic] ((((((((((((((((

Level 1

27. Find sin ( and tan ( if [pic] where 90( ( ( ( 180(.

28. Find cos ( and tan ( if [pic] where ( lies in quadrant IV.

29. Find sin ( and cos ( if [pic] where ( lies in quadrant III.

Level 2

30. If [pic] and 90( ( ( ( 180(, find the value of [pic].

31. If [pic] and 180( ( ( ( 270(, find the value of [pic].

32. If [pic] and 270( ( ( ( 360(, find the value of [pic].

33. If [pic] and sin ( ( 0, find the value of [pic].

34. If tan ( ( (k where 90( ( ( ( 180(, express sin ( and cos ( in terms of k.

35. It is given that [pic] where ( lies in quadrant IV.

(a) find tan ( in terms of k.

(b) find sin ( in terms of k.

(c) Hence find [pic] in terms of k.

36. If [pic] where c ( b ( 0 and ( lies in quadrant III,

(a) express sin ( and tan ( in terms of b and c.

(b) find the value of [pic].

37. A(1, 0) and B(x, y) are two points on the unit circle with centre at the origin and (AOB ( (. If y ( (4x, find the possible values of sin (, cos ( and tan (.

38. It is given that [pic] and 90( ( ( ( 180(.

(a) Find the value of tan (.

(b) Hence find (. (Correct your answer to 1 decimal place.)

39. If [pic], tan B ( 3 where A and B lie in quadrants IV and III respectively, find the values of the following.

(a) cos A sin B

(b) tan A tan B ( cos A cos B

40. It is given that sin ( ( p and tan ( ( 2p where p ( 0.

(a) In which quadrant does the angle ( lie?

(b) Find cos (.

(c) Find (.

(d) Hence find the value of p.

Exercise 7D

((((((((((((((( Elementary Set [pic] (((((((((((((((

Level 1

1. Express the following expressions in terms of sin (.

(a) [pic] (b) [pic]

2. Express the following expressions in terms of cos (.

(a) [pic] (b) [pic]

3. Express the following expressions in terms of tan (.

(a) [pic] (b) [pic]

Simplify the following expressions. (4 ( 11)

4. sin(180( ( () 5. cos(180( ( () 6. tan(360( ( ()

7. cos(90( ( 2() 8. sin(270( ( 2() 9. tan(90( ( 2()

10. [pic] 11. [pic]

12. Find the value of cos 330( ( sin150( without using a calculator.

Prove the following identities. (13 – 14)

13. [pic] 14. cos(90( ( () tan(270( ( () ( (cos (

Level 2

15. Express [pic] in terms of sin (.

Simplify the following expressions. (16 – 17)

16. a cos(180( ( () ( a sin(180( ( () 17. [pic]

((((((((((((((( Intermediate Set [pic] (((((((((((((((

Level 1

Simplify the following expressions. (18 ( 24)

18. [pic] 19. [pic] 20. [pic]

21. [pic] 22. [pic] 23. [pic]

24. [pic]

Prove the following identities. (25 – 26)

25. [pic] 26. [pic]

Level 2

27. Express [pic] in terms of sin (.

28. Find the value of [pic] without using a calculator.

Simplify the following expressions. (29 – 30)

29. [pic]

30. sin(360( ( () cos(90( ( () ( sin(270( ( () cos(180( ( ()

Prove the following identities. (31 – 33)

31. [pic]

32. [pic]

33. [pic]

34. If [pic] and A lies between 180( and 270(, find the values of the following.

(a) [pic]

(b) sin(270( ( A)

(c) [pic]

(((((((((((((((( Advanced Set [pic] ((((((((((((((((

Level 1

Simplify the following expressions. (35 ( 37)

35. [pic] 36. cos(2( ( 180()

37. cos2(450( ( ()

Prove the following identities. (38 ( 39)

38. [pic] 39. [pic]

Level 2

Simplify the following expressions. (40 ( 42)

40. [pic] 41. [pic]

42. [pic]

43. Find the value of [pic] without using a calculator.

44. Express [pic] in terms of cos (.

Prove the following identities. (45 ( 48)

45. [pic]

46. [pic]

47. [pic]

48. [pic]

49. Express the following expressions in terms of sin (.

(a) [pic] (b) [pic]

50. Express the following expressions in terms of cos (.

(a) [pic] (b) [pic]

51. Let [pic] and 180( ( x ( 225(.

(a) Find the values of the following expressions.

(i) sin x ( cos x (ii) sin x ( cos x

(b) Hence find sin x and x.

52. Suppose sin ( cos ( ( 2,

(a) find the value of (sin ( ( cos ()2.

(b) Can an angle ( lying between 0( and 360( which satisfies sin ( cos ( ( 2 be found? Explain briefly.

Exercise 7E

((((((((((((((( Elementary Set [pic] (((((((((((((((

Level 1

1. The figure shows the graph of y ( sin x for 0( ( x ( 360(.

[pic]

(a) Find the value of sin 295( from the graph.

(b) If sin x ( (0.8, find x from the graph.

(c) Find the range of x from the graph such that sin x ( 0.7.

2. The figure shows the graph of y ( cos x for 0( ( x ( 360(.

[pic]

(a) Find the value of cos 250( from the graph.

(b) If cos x ( 0.6, find x from the graph.

(c) Find the range of x from the graph such that cos x ( (0.5.

Find the maximum and minimum values of the following expressions for 0( ( ( ( 360(. (3 – 8)

3. 2 sin ( 4. 5 cos ( 5. 4 ( 2 sin (

6. 2 ( 4 cos ( 7. 3 ( 2 cos ( 8. 3 sin 2(

Level 2

Sketch the graphs of the following trigonometric functions for 0( ( x ( 360(. (9 ( 11)

9. (a) y ( 2 sin x (b) [pic] (c) [pic]

10. (a) y ( 2 cos x (b) y ( 2 cos(x ( 45() (c) y ( 2 cos 3x

11. (a) y ( 3 tan x (b) y ( tan(x ( 45() (c) y ( tan(x ( 90() ( 2

12. The figure shows the graph of y ( A cos x ( B for 0( ( x ( 360(.

[pic]

(a) From the graph, find the period and amplitude.

(b) Find the values of A and B.

(c) Find the range of values of y such that 0(  ................
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