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Chapter 7 Trigonometry Chapter 7 Trigonometry WARM - UP E XERCISE 1. Simplify the following. (a) (b) 3 24 2. Rationalize the following. 1 (a) 2 (b) 5 36 4 5 15 3. Which of the following is a right-angled triangle? Explain your answer. A 5 (a) C (b) A 8 4 B 7 12 B 4 C 4. Find the values of x in the following figures. (a) B (b) B x x 5 A A 4 9 D 12 25 C C 5. Simplify the following. (a) 2 sin 2 2 sin 2 (90 ) 6. Simplify the following. tan (a) 1 tan 2 (b) tan(90 ) cos (b) (sin cos ) 2 (cos sin ) 2 1 2 New Trend Mathematics S4B — Supplement B UILD - UP E XERCISE [ 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 El em en tar y S et Level 1 1. In each of the following figures, find . (Correct your answers to 1 decimal place.) (a) (b) 5 7 8 4 (c) 3 1 In each of the following, find sin , cos and tan . Leave your answers in surd form if necessary. (2 5) 2. 3. 8 5 5 4 Ex.7A Elementary Set y 4. y 5. P 20 (24, 7) O x O 6. Find the value of sin 30 cos 60. 7. If cos 2 where 0 90, find sin and tan . 3 8. If sin 4 where 0 90, find cos and tan . 5 9. If tan 2 where 0 90, find sin and cos . Level 2 10. If cos 12 where 0 90, find sin tan . 13 16 x 3 Chapter 7 Trigonometry 1 2 where 0 90. 12. Find such that cos( 10) 1 where 0 90. (Correct your answer to 1 decimal place.) 3 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.) D Ex.7A Elementary Set 11. Find such that sin C 6 cm A 85 9 cm B Intermediate Set Level 1 14. In the figure, find . (Correct your answer to 1 decimal place.) 17 In each of the following, find sin , cos and tan . Leave your answers in surd form if necessary. (15 18) 15. 16. y 5 (3, 7) 10 x O y 17. 18. 4 12 P 25 O 15 x 5 Ex.7A Intermediate Set 15 4 New Trend Mathematics S4B — Supplement 19. In the figure, find the unknowns. (Correct your answers to 1 decimal place.) 40 12 x 6 20. If sin 5 8 7 where 0 90, find cos and tan . 21. Find the value of tan 2 45 cos 60 . (Leave your answer in surd form.) sin 45 Level 2 22. If cos 3 10 where 0 90, find 2 tan 2 sin 2 cos2 . 23. If tan 2 where 0 90, find sin 2 2 cos 2 . Ex.7A Intermediate Set 24. Find such that cos(90 ) 3 where 0 90. 2 25. Find such that tan( 20) 2 where 0 90. (Correct your answer to 1 decimal place.) 3 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.) C D 5 cm A 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.) F 8 cm 120 cm 60 A B 40 cm B 25 C D 80 cm E 5 Chapter 7 Trigonometry Advanced Set Level 1 In each of the following, find sin , cos and tan . Leave your answers in surd form if necessary. (28 31) y 28. A 29. P (5, 11) 3 B C 7 x O y 30. 31. BCD is a straight line. A P O 14 12 x 8 B 45 D 6 C A 32. In the figure, BCD is a straight line. Find the unknowns. (Correct your answers to 1 decimal place.) Ex.7A Advanced Set y 40 x 20 B C D 7 33. Find the value of 3 (cos 2 45 4 sin 2 45) tan 45 2 sin 60 . (Leave your answer in surd form.) Level 2 34. If tan 2 3 5 where 0 90, find 2 sin cos . (Leave your answer in surd form.) 35. If tan 3 where 0 90, find 2 sin cos . sin 3 cos 36. Find such that sin 3 1 where 0 90. 1 37. Find such that tan( 15) where 0 90. 2 3 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.) F 180 m A 30 B 150 m 20 D C xm 200 m E 6 New Trend Mathematics S4B — Supplement 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.) A F 20 cm 50 B C D E 15 cm 200 m A Ex.7A Advanced Set 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.) 25 cm B 40 50 hm hm C 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.) B 25 cm 15 cm C 15 cm 15 cm D 30 A Exercise 7B [ In this exercise, leave your answers in surd form if necessary. ] Ex.7B Elementary Set El em en tar y S et Level 1 1. In each of the following figures, find . y (a) (b) y 230 145 x x Chapter 7 Trigonometry 7 y (c) x 30 2. Find the reference angle corresponding to each of the following angles. (a) 123 (b) 217 (c) 297 4. Find sin , cos and tan in the figure. y x O Ex.7B Elementary Set 3. If sin 0 and tan 0, determine the quadrant in which lies. (6, 4) 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) Intermediate Set 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. Ex.7B Intermediate Set Level 2 Find the value of each of the following expressions. (10 12) 10. sin 45cos120 11. 2 sin 135 3 cos 225 12. 2 (cos 330 sin 150) 8 New Trend Mathematics S4B — Supplement 15. Find sin , cos and tan in the figure. y (7, 3) x O Ex.7B Intermediate Set 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. 1 1 sin 225 cos 315 3 4 1 24. tan(225) 4 sin 135 2 25. cos 120 sin 225 sin 330 cos 225 Advanced Set Level 1 sin 2 0 , determine the quadrant(s) in which lies. 26. If tan Ex.7B Advanced Set 27. Find sin , cos and tan in the figure. y x O (5, 5) 28. Express cos (120) in terms of acute angle. Chapter 7 Trigonometry 9 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) 32. cos (210) 31. tan 405 34. 3 cos (60) 4 tan (225) 33. sin (30) cos (120) 35. 2 36. cos150 sin 240 3 sin 120 tan150 5 sin 240 3 37. In the figure, AOC is a straight line. ADB is a straight line perpendicular to the x-axis and AB BC. Find sin . Ex.7B Advanced Set Level 2 Find the value of each of the following expressions. (33 36) y A (3, 4) D x O B C (3, 4) Exercise 7C [ In this exercise, leave your answers in surd form if necessary. ] El em en tar y S et Level 1 1. If sin 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 cos 2 where 90 180. 3 5. Find cos and tan if sin 1 where 270 360. 3 6. Find sin and cos if tan 1 where 180 270. 3 7. Find sin and cos if tan 3 where lies in quadrant III. 4 Ex.7C Elementary Set 2. If cos 0, find the range of where 0 360. 10 New Trend Mathematics S4B — Supplement 8. Find sin and tan if cos 12 where lies in quadrant II. 13 Ex.7C Elementary Set 3 9. Find cos and tan where lies in quadrant IV if sin . 5 Level 2 10. If tan 5 and 180 270, find the value of 3 sin 2 cos . 12 11. If sin 1 and 90 180, find the value of sin tan . 3 12. If cos 1 and 270 360, find the value of 4 tan sin . 2 13. If tan 1 and cos 0, find sin . Intermediate Set Level 1 1 14. Find sin and tan if cos where 270 360. 2 Ex.7C Intermediate Set 15. Find cos and tan if sin 1 where 180 270. 2 16. Find sin and cos if tan 1 where 90 180. 3 17. Find sin and cos if tan 5 where lies in quadrant IV. 2 18. Find sin and tan if cos 3 where lies in quadrant III. 7 19. Find cos and tan if sin Level 2 20. If sin 1 where lies in quadrant II. 3 5 sin 2 cos and 270 360, find the value of . 2 sin cos 13 21. If tan 1 sin 2 cos and 90 180, find the value of . 2 2 sin 3 cos 22. If cos 2 3 sin and 180 270, find the value of 4 tan . 3 cos 23. If sin 1 5 and tan 0, find the value of 5 (sin cos ) . 11 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. 1 (c) If tan and 180 270, express sin and cos in terms of k. k 5 3 25. If cos A , cos B and both A and B are obtuse angles, find the values of the following. 6 5 (a) cos Acos B (b) 2 sin A cos B 2 sin B cos A tan A tan B (c) 1 tan A tan B Ex.7C Intermediate Set Chapter 7 Trigonometry 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 Level 1 2 27. Find sin and tan if cos where 90 180. 5 28. Find cos and tan if sin 29. Find sin and cos if tan 3 17 where lies in quadrant IV. 3 where lies in quadrant III. 2 30. If tan 1 3 2 tan 2 . and 90 180, find the value of 2 2 cos 31. If sin 2 32. If cos 33. If cos 13 and 180 270, find the value of 2 cos 2 tan . 2 3 tan 2 1 and 270 360, find the value of . 2 sin 3 1 and sin 0, find the value of tan . 10 tan 34. If tan k where 90 180, express sin and cos in terms of k. 1 where lies in quadrant IV. k (a) find tan in terms of k. (b) find sin in terms of k. 2 (c) Hence find tan 2 in terms of k. sin 2 35. It is given that cos Ex.7C Advanced Set Level 2 12 New Trend Mathematics S4B — Supplement b where c b 0 and lies in quadrant III, c (a) express sin and tan in terms of b and c. 3 (b) find the value of 3 tan 2 . cos 2 36. If cos 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 . Ex.7C Advanced Set 2 sin cos 1 and 90 180. cos 3 sin 4 (a) Find the value of tan . (b) Hence find . (Correct your answer to 1 decimal place.) 38. It is given that 1 39. If sin A , tan B 3 where A and B lie in quadrants IV and III respectively, find the values of 3 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 El em en tar y S et Level 1 1. Express the following expressions in terms of sin . Ex.7D Elementary Set (a) cos tan (b) cos2 2. Express the following expressions in terms of cos . (a) sin 2 tan (b) sin 2 1 3. Express the following expressions in terms of tan . sin cos (a) 4 tan (b) cos sin Chapter 7 Trigonometry 13 4. sin(180 ) 5. cos(180 ) 6. tan(360 ) 7. cos(90 2) 8. sin(270 2) 9. tan(90 2) 10. sin( 90 ) sin( 90 ) 11. cos(90 ) cos(90 ) 12. Find the value of cos 330 sin150 without using a calculator. Prove the following identities. (13 – 14) 13. sin 2 (90 ) tan 2 (180 ) sin 2 Level 2 15. Express 14. cos(90 ) tan(270 ) cos Ex.7D Elementary Set Simplify the following expressions. (4 11) cos(90 ) cos() in terms of sin . cos(180 ) Simplify the following expressions. (16 – 17) 16. a cos(180 ) a sin(180 ) 17. cos2 (270 ) sin( 90 ) cos(180 ) 21. tan(180 ) sin( 270 ) 24. 1 tan 2 (90 ) cos (90 ) 22. sin( 90 3 ) cos(270 ) 3 20. cos(270 ) 2 23. tan 2 1 cos2 2 Prove the following identities. (25 – 26) sin(180 ) 2 cos 25. cos() tan(360 ) 26. cos(180 ) tan(180 ) sin( ) sin( 90 ) cos(180 ) Level 2 27. Express 3sin(180 ) cos(90 ) 4 sin( 270 ) cos(360 ) in terms of sin . 28. Find the value of tan 240 cos150 3 sin 120 without using a calculator. cos 240 Ex.7D Intermediate Set Intermediate Set Level 1 Simplify the following expressions. (18 24) 18. sin( 270 ) 19. tan(270 ) 2 2 14 New Trend Mathematics S4B — Supplement Simplify the following expressions. (29 – 30) ab ab 29. 2 2 sin (180 ) cos (180 ) 30. sin(360 ) cos(90 ) sin(270 ) cos(180 ) Ex.7D Intermediate Set Prove the following identities. (31 – 33) sin( 90 ) sin 2 0 31. cos(90 ) tan(270 ) 32. 2 tan(90 ) tan(360 ) 33. 1 3 cos 2 sin cos sin( 90 ) cos(360 ) sin 1 cos(180 ) tan(270 ) 34. If sin 2 A 64 and A lies between 180 and 270, find the values of the following. 289 (a) cos A (b) sin(270 A) (c) sin 2 (90 A) cos(270 A) tan 2 (270 A) Advanced Set Level 1 Simplify the following expressions. (35 37) 35. tan( 180 ) 36. cos(2 180) 3 Ex.7D Advanced Set 37. cos 2(450 ) Prove the following identities. (38 39) sin( ) tan 38. sin( 270 ) 39. sin( 90 ) tan() 1 sin(180 ) Level 2 Simplify the following expressions. (40 42) 40. cos(180 ) tan(270 ) sin( 90 ) 41. 2 tan(180 ) sin(180 ) cos(360 ) Chapter 7 Trigonometry 42. 15 (a b) tan(270 ) (a b) tan(180 ) tan(90 ) tan 43. Find the value of 44. Express cos 40 tan140 sin 220 without using a calculator. sin 40 tan 220 cos140 cos()[1 sin 2 (180 )] 1 cos 2 (180 ) sin 2 () in terms of cos . Prove the following identities. (45 48) 1 45. tan(180 ) tan(270 ) sin cos cos() 0 sin( 180 ) 47. 1 1 2 tan 1 sin 1 sin cos 48. sin(180 ) 3 tan(180 ) tan() tan(360 ) cos(360 ) 49. Express the following expressions in terms of sin . 1 1 (a) (b) (cos2 sin 2 ) cos 2 1 cos 1 cos 50. Express the following expressions in terms of cos . (a) sin tan (b) sin 2 tan2 1 and 180 x 225. 4 (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. 51. Let sin x cos 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. Ex.7D Advanced Set 46. tan(270 ) 16 New Trend Mathematics S4B — Supplement Exercise 7E El em en tar y S et Level 1 1. The figure shows the graph of y sin x for 0 x 360. y 1 y sin x x O 90 180 270 360 1 (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. Ex.7E Elementary Set 2. The figure shows the graph of y cos x for 0 x 360. y 1 y cos x O x 90 180 270 360 1 (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) 1 1 9. (a) y 2 sin x (b) y sin x (c) y 2 sin x 2 2 Chapter 7 Trigonometry 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 17 12. The figure shows the graph of y A cos x B for 0 x 360. y y A cos x B 2 O 90 180 270 360 x 2 4 6 (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 < x < 90. Ex.7E Elementary Set Find the maximum and minimum values of the following expressions for 0 x 360. (13 – 14) 13. 4 cos2 x 14. (3 sin x 1) 2 15. The figure shows the graphs of y sin x and y cos x for 0 x 360. y 1 y sin x O x 30 60 90 120 150 180 210 y cos x 1 (a) Find the values of the following from the graph. (i) sin 156 (ii) cos 162 (b) Find x from the graph such that cos x sin x. (c) Find the range of x from the graph such that cos x sin x. 240 270 300 330 360 18 New Trend Mathematics S4B — Supplement Intermediate Set Level 1 16. The figure shows the graph of y sin 2x for 0 x 180. y 1 y sin 2x x O 45 90 135 180 1 (a) Find the value of sin 2(120) from the graph. (b) If sin 2x 0.4, find x from the graph. (c) Find the range of x from the graph such that sin 2x 0.8. Ex.7E Intermediate Set 17. The figure shows the graph of y cos y 1 x for 0 x 720. 2 x y cos 2 x O 180 360 540 720 1 (a) Find the value of cos (b) If cos 100 from the graph. 2 x 0.7 , find x from the graph. 2 x (c) Find the range of x from the graph such that 0.5 cos 0 . 2 Find the maximum and minimum values of the following expressions for 0 360. (18 – 20) 1 5 18. 4 sin 5 19. cos 3 20. 2 sin 2 2 Level 2 Sketch the graphs of the following trigonometric functions for 0 x 360. (21 23) 21. (a) y 2 sin x (b) y 2 sin x 1 (c) y 2 sin x 2 22. (a) y 3 cos x (b) y 3 cos(x 45) (c) y 3 cos 3x Chapter 7 Trigonometry 23. (a) y 2 tan x (b) y tan(x 45) 19 (c) y 2 tan(x 90) 1 24. The figure shows the graph of y A cos x B for 0 x 360. y y A cos x B 3 2 1 x O 90 180 270 360 1 (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 x lies in quadrant II. 25. (a) 3sin 2 x 1 (b) 4 cos 2 x 6 26. (a) (cos x 2) 2 1 (b) ( sin x 3) 2 3 27. (a) (sin x 1) 2 2 (b) (2 cos x 1) 2 2 28. (a) 1 4 cos x 5 (b) Ex.7E Intermediate Set Find the maximum and minimum values of the following expressions for 0 x 360. (25 – 28) 1 4 sin 2 x 29. The figure shows the graphs of y sin 2x and y cos x for 0 x 360. y 1 y sin 2x O y cos x x 30 60 90 120 150 180 210 240 1 (a) Find the values of the following from the graph. (i) sin 246 (ii) cos 216 (b) Find x from the graph such that sin 2x cos x. (c) Find the range of x from the graph such that cos x sin 2x. 270 300 330 360 20 New Trend Mathematics S4B — Supplement Advanced Set Level 1 30. The figure shows the graph of y cos 2x for 0 x 180. y 1 y cos 2x x O 45 90 135 180 1 (a) Find the value of cos 110 from the graph. (b) If cos 2x 0.8, find x from the graph. (c) Find the range of x from the graph such that cos 2x 0.5. 31. The figure shows the graph of y sin Ex.7E Advanced Set x for 0 x 720. 2 y 1 O x y sin 2 x 180 360 540 720 1 (a) Find the value of sin 40 from the graph. x (b) If sin 0.4 , find x from the graph. 2 (c) Find the range of x from the graph such that sin x 0.8 . 2 Level 2 Sketch the graphs of the following trigonometric functions for 0 x 360. (32 34) 1 1 1 1 1 32. (a) y sin x (b) y sin x (c) y sin x 2 2 2 2 2 33. (a) y cos x (b) y cos(x 45) (c) y cos 3x 34. (a) y 2 tan x 4 (b) y 3 tan(x 45) (c) y tan(x 90) 3 Chapter 7 Trigonometry 21 35. The figure shows the graph of y A cos x B for 0 x 360. y y A cos x B 8 6 4 2 O 90 180 270 x 360 (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 x lies in quadrant IV. 36. (a) 3 4 sin 2 x (b) 3 4 cos3 x 37. (a) (5 cos x 1) 2 (b) (2 sin x 1) 2 38. (a) 3 (cos 2 x 5) (b) 2 39. (a) ( 2 cos x 3) 2 (b) Ex.7E Advanced Set Find the maximum and minimum values of the following expressions for 0 x 360. (36 – 39) 1 (sin 3) 2 x 2 2 4 2 sin 2 x 40. By letting y sin x and expressing the following expressions in the form of a(y b) 2 c where a, b and c are integers, find the maximum and minimum values of the following expressions for 0 x 360. (a) 2 sin 2x 4 sin x 1 (b) 13 12 sin x 4 cos 2x 41. The figure shows the graphs of y sin x and y cos x for 0 x 360. 2 y 1 y sin x O 1 x 30 60 90 120 150 180 210 240 270 x y cos 2 300 330 360 22 New Trend Mathematics S4B — Supplement Ex.7E Advanced Set (a) Find the values of the following from the graph. (i) sin 138 (ii) cos 117 x (b) Find x from the graph such that sin x cos . 2 x (c) Find the range of x from the graph such that cos sin x . 2 3 1 42. For 0 x 360, the maximum and minimum values of A B sin x are and respectively 2 2 where A and B are positive numbers. (a) Find the values of A and B. (b) Explain briefly whether the maximum and minimum values of A B sin x for 0 x 360 have the same values as those of A B sin x. Exercise 7F [ In this exercise, correct your answers to 1 decimal place if necessary. ] El em en tar y S et Level 1 Solve the following equations for 0 360. (1 – 15) 1. tan tan 15 2. sin sin 24 4. sin sin 123 7. sin 2 2 8. cos Ex.7F Elementary Set 10. sin 2 0 13. sin( 10) 5. cos cos 100 1 2 1 2 3. cos cos 79 6. tan tan 111 9. tan 3 11. sin 0.6 12. cos 2 0.1 14. 2 sin 1 15. sin cos 0 Level 2 Solve the following equations for 0 360. (16 – 22) 16. 3 sin tan 17. cos 2 cos 0 18. 2 sin 2 sin 3 0 19. 2 sin 2 cos 2 0 20. cos tan sin 21. sin 2 2 sin cos cos 2 0 22. sin 2 1 4 cos 2 sin 3 in the form of tan k. sin cos 4 cos 2 sin 3 (b) Hence solve for 0 360. sin cos 4 23. (a) Rewrite the equation Chapter 7 Trigonometry Intermediate Set Level 1 Solve the following equations for 0 360. (24 – 35) 24. cos 50 cos 25. sin sin 165 27. cos 30. sin 2 2 1 4 2 33. tan( 60) 5 23 26. tan 195 tan 3 2 28. tan 4 29. sin 31. cos( 40) 0.2 32. tan( 35) 34. 3 cos 2 35. 1 3 1 cos 2 sin 0 3 36. 2 sin 3 tan 0 37. 2 sin 2 3 sin 38. 3 tan 2 1 39. 2 tan 2 3 tan 1 0 40. 2 cos 2 5 sin 1 0 41. 2 cos 3 tan 42. 2 cos 2 3 cos sin 2 sin 2 0 Ex.7F Intermediate Set Level 2 Solve the following equations for 0 360. (36 – 42) 4 sin 1 in the form of tan k. sin 3 cos 2 4 sin 1 (b) Hence solve for 0 360. sin 3 cos 2 43. (a) Rewrite the equation 44. (a) Rewrite the equation 5 tan 2 cos 0 in the form of a sin 2 b sin c 0, where a, b and c are integers. (b) Hence solve 5 tan 2 cos 0 for 0 360. cos 1 , sin 2 3 (a) rewrite the equation in the form of a cos 2 b cos c 0 where a, b and c are integers. cos 1 (b) Hence solve for 0 360. sin 2 3 Advanced Set Level 1 Solve the following equations for 0 360. (46 – 51) 1 46. tan 47. sin sin 46 2 3 Ex.3A Advanced Set 45. Suppose 24 New Trend Mathematics S4B — Supplement 48. cos 3 2 3 4 5 49. tan 50. 3sin 2 3 2 3 51. cos( 50) 3 2 Level 2 Solve the following equations for 0 360. (52 – 59) 52. (tan 1)sin 0 54. 3 sin 2 cos 2 56. sin 2 sin 0 2 2 58. 2 cos 2 3 sin 3 53. 2 cos2 2 55. 2 sin tan 2 sin 0 57. 4sin 2 2sin 2 59. 3 cos 2 4 sin cos 4 sin 2 0 Ex.7F Advanced Set 60. (a) Rewrite the equation 4 tan 3 cos 0 in the form of a sin 2 b sin c 0, where a, b and c are integers. (b) Hence solve 4 tan 3 cos 0 for 0 360. 1 5 cos , where 0 180. sin (a) Rewrite the above equation in the form of a cos 2 b cos c 0, where a, b and c are integers. 1 5 cos (b) Hence solve tan . sin 61. It is given that tan sin 3 , cos 8 (a) rewrite the equation in the form of a sin 2 b sin c 0. sin 3 for 0 360. (b) Hence solve 2 cos 8 62. Suppose 2 63. Solve the equation 4 sin 2 3 sin cos 2 for 0 360. 64. Solve the equation 2 sin 2 cos (tan 4 cos ) 0 for 0 360. 65. (a) Expand (x 1) 4. (b) Hence solve cos 4 4 cos 3 6 cos 2 4 cos 1 0 for 0 360. Chapter 7 Trigonometry Exercise 7G El em en tar y S et Level 1 1. The figure shows the graph of y 2 sin x for 0 x 360. 25 y y 2 sin x 2 1 x O 30 60 90 120 150 180 210 240 270 300 330 360 300 330 360 1 2 Solve the equation 2 sin x 0 graphically. 2. The figure shows the graph of y cos 2x 1 for 0 x 360. y cos 2x 1 2 1 x O 30 60 90 120 150 180 210 240 270 Solve the equation cos 2x 1 0 graphically. 3. The figure shows the graph of y cos(x 60) for 0 x 360. y 1 y cos (x 60) O x 30 60 90 120 150 180 210 1 Solve the equation cos(x 60) 0 graphically. 240 270 300 330 360 Ex.7G Elementary Set y 26 New Trend Mathematics S4B — Supplement 4. The figure shows the graph of y 4 cos 2x for 0 x 180. y 4 y 4 cos 2x 2 x O 30 60 90 120 150 180 2 4 Solve the equation 4 cos 2x 0 graphically. Ex.7G Elementary Set 5. The figure shows the graph of y 1 2 sin(x 30) for 0 x 360. y 3 y 1 2 sin (x 30) 2 1 O x 30 60 90 120 150 180 210 1 Solve the equation 1 2 sin(x 30) 0 graphically. 240 270 300 330 360 Chapter 7 Trigonometry 27 Level 2 6. The figure shows the graph of y 2 tan x sinx for 0 x 360. y y 2 tan x sin x 20 10 x O 30 60 90 120 150 180 210 240 270 300 330 360 10 Solve the following equations graphically. (a) 2 tan x sin x 20 (b) 2 tan x sin x 15 0 (c) 4 tan x 2 sin x 36 0 7. The figure shows the graph of y a sin x 2 cos x for 0 x 180. y 2 y a sin x 2 cos x 1 O x 30 60 90 120 150 180 1 2 (a) Find the value of a. (b) Solve the equation a sin x 2 cos x 1 0 graphically. Ex.7G Elementary Set 20 28 New Trend Mathematics S4B — Supplement 8. The figure shows the graph of y a sin x b cos x for 0 x 180. y 2 y a sin x b cos x Ex.7G Elementary Set 1 x O 30 60 90 120 150 180 1 2 (a) Find the values of a and b. (b) Solve the equation 4 sin x 3 cos x 2 graphically. Intermediate Set Level 1 1 9. The figure shows the graph of y sin x for 0 x 360. 2 y 0.5 Ex.7G Intermediate Set O y 1 sin x 2 120 150 x 30 60 90 180 0.5 Solve the equation 1 sin x 0 graphically. 2 210 240 270 300 330 360 Chapter 7 Trigonometry 29 10. The figure shows the graph of y 2 cos 2x 1 for 0 x 360. y 3 y 2 cos 2x 1 2 1 x O 30 60 90 120 150 180 210 240 270 300 330 360 1 Solve the equation 2 cos 2x 1 0 graphically. 11. The figure shows the graph of y sin(x 30) for 0 x 360. y 1 x O 30 60 90 120 150 180 210 240 270 300 330 360 330 360 1 Solve the equation sin(x 30) 0 graphically. Level 2 12. The figure shows the graph of y 2 3 cos(x 20) for 0 x 360. y 5 y 2 3 cos (x 20) 4 3 2 1 O 1 x 30 60 90 120 150 180 210 240 270 300 Ex.7G Intermediate Set y sin (x 30) 30 New Trend Mathematics S4B — Supplement Solve the following equations graphically. 2 1 (a) cos(x 20) (b) cos(x 20) 3 3 (c) 3 cos(x 20) 1 13. The figure shows the graph of y 3 tan x sin x for 0 x 360. y y 3 tan x sin x 40 30 20 10 x O 30 60 90 120 150 180 210 240 270 300 330 360 10 20 Ex.7G Intermediate Set 30 40 Solve the following equations graphically. (a) 3 tan x sin x 30 (b) 3 tan x sin x 10 0 14. The figure shows the graph of y a sin bx for 0 x 180. y y a sin bx 4 2 O 2 4 x 30 60 90 120 150 180 (c) 6 tan x 12 2 sin x Chapter 7 Trigonometry 31 (a) Find the values of a and b. (b) Solve the equation a sin bx 2 graphically. (c) Solve the equation 2 sin bx 1 0 graphically. 15. The figure shows the graph of y a sin x b cos x for 0 x 360. y y a sin x b cos x 2 1 O x 30 60 90 120 150 180 210 240 270 300 330 360 1 2 (b) Solve the equation a sin x b cos x 2 0 graphically. 14 (c) Solve the equation 2(a sin x b cos x) 0 graphically. 5 16. The figure shows the graph of y A tan x B for 0 x 180. y 6 4 2 y A tan x B O x 30 60 90 120 150 180 2 4 6 (a) Find the values of A and B. (b) Solve the following equations graphically. (i) A tan x B 2 (ii) tan x 2 Ex.7G Intermediate Set (a) Find the values of a and b. 32 New Trend Mathematics S4B — Supplement Advanced Set Level 1 17. The figure shows the graph of y 4 sin 2x for 0 x 360. y 4 y 4 sin2 x 2 x O 30 60 90 120 150 180 210 240 270 300 330 360 300 330 360 Solve the equation 4 sin 2x 0 graphically. Level 2 18. The figure shows the graph of y 2 3 sin 2x for 0 x 360. y Ex.7G Advanced Set 5 y 2 3 sin 2x 4 3 2 1 O x 30 60 90 120 150 180 1 Solve the following equations graphically. (a) 3 sin 2x 2 (b) 3 sin 2x 1 (c) sin 2x 0.6 210 240 270 Chapter 7 Trigonometry 33 19. The figure shows the graph of y 2 cos(x 45) for 0 x 360. y 2 y 2cos (x 45) 1 x O 30 60 90 120 150 180 210 240 270 300 330 360 1 2 Solve the following equations graphically. (a) cos(x 45) 0 (b) cos(x 45) 0.8 (c) 4cos(x 45) 3.2 0 20. The figure shows the graph of y 3 2 sin(x 60) for 0 x 360. 5 Ex.7G Advanced Set y y 3 2 sin (x 60) 4 3 2 1 O x 30 60 90 120 150 180 Solve the following equations graphically. (a) sin(x 60) 0 (b) sin(x 60) 0.5 (c) 3 sin(x 60) 5 210 240 270 300 330 360 34 New Trend Mathematics S4B — Supplement 21. The figure shows the graph of y 2 tan x 10 cos x for 0 x 360. y y 2 tan x 10 cos x 40 30 20 10 x O 30 60 90 120 150 180 210 240 270 300 330 360 300 330 360 10 20 30 40 Ex.7G Advanced Set Solve the following equations graphically. (a) 2 tan x 10 cos x 6 (b) 2 tan x 10 cos x 18 0 (c) tan x 10 5 cos x 22. The figure shows the graph of y a cos bx 2 for 0 x 360. y 5 y a cos bx 2 4 3 2 1 O x 30 60 90 120 150 180 210 1 (a) Find the values of a and b. (b) Solve the equation a cos bx 2 graphically. (c) Solve the equation 5a cos bx 2 0 graphically. 240 270 Chapter 7 Trigonometry 35 23. The figure shows the graph of y a sin x b cos x for 0 x 360. y y a sin x b cos x 4 3 2 1 x O 30 60 90 120 150 180 210 240 270 300 330 360 1 2 3 4 Ex.7G Advanced Set (a) Find the values of a and b. (b) Solve the equation a sin x b cos x 2 0 graphically. a sin x b cos x (c) Solve the equation 1 0 graphically. 3 24. The figure shows the graph of y 3 sin x a cos x for 0 x 360. y 4 y 3 sin x a cos x 3 2 1 O x 30 60 90 120 150 180 210 240 1 2 3 4 (a) Find the values of a. (b) Solve the equation 3 sin x a cos x 3 0 graphically. (c) Find the range of x for which 3 sin x a cos x 5 0. 270 300 330 360 36 New Trend Mathematics S4B — Supplement 25. The figure shows the graph of y a sin(x b) for 0 x 180. y 4 y a sin (x b) 3 Ex.7G Advanced Set 2 1 x O 30 60 90 120 150 180 1 2 (a) Find the values of a and b. b (b) Solve a sin( x b) graphically. 40 C HAPTER T EST (Time allowed: 1 hour) Section A 1. In the figure, find sin , cos and tan . (Leave your answers in surd form if necessary.) (3 marks) 3 4 2. In the figure, find sin , cos and tan . (Leave your answers in surd form if necessary.) (3 marks) y O (4, 2) x Chapter 7 Trigonometry 3. Solve the equation 37 2 sin 3 cos for 0 x 360. (Correct your answers to 1 decimal place.) (3 marks) 3 where 90 180, find cos and tan . (Leave your answers in surd form if 2 necessary. ) (4 marks) 4. If sin 5. If tan 1, find the value of 6. Find the value of cos 2 (180 ) . tan() (4 marks) 1 1 tan 240 sin 120 cos 300 . (Leave your answer in surd form.) 3 2 (4 marks) 7. Find the maximum and minimum values of the following expressions for 0 x 360. (a) 1 sin x (2 marks) 1 (b) (2 marks) 3 2 cos x Section B 8. (a) Rewrite 2 cos 3 tan in the form of a sin 2 b sin c 0, where a, b and c are integers. (2 marks) (b) Hence solve 2 cos 3 tan for 0 360. (2 marks) (c) Let f() 2 cos 2 3 tan cos . (i) Rewrite f () in the form of a(sin h) 2 + k, where a, h and k are constants. (ii) Find the maximum and minimum values of f () for 0 360. (6 marks) 3 9. The figure shows the graph of y sin x cos x for 0 x 180. 2 y 2 y sin x 3 cos x 2 1 O x 30 60 90 120 150 180 1 2 3 (a) Find the maximum and minimum values of y sin x cos x from the graph for 0 x 180. 2 (2 marks) 38 New Trend Mathematics S4B — Supplement (b) Solve the following equations graphically. 3 (i) sin x cos x 1 0 2 3 (ii) sin x cos x 1.4 0 2 (iii) 2 sin x 3 cos x 3 0 (3 marks) 3 (c) Find the range of values of k such that the equation sin x cos x k 0 does not have 2 solutions for 0 x 180. (5 marks) Multiple Choice Questions (3 marks each) 10. In the figure, sin 10 . cos 50 10 sin 25 B. . cos 50 C. 10 cos 25 sin 50 . 10 10 D. . tan 25 tan 50 A. 2 2 8 A. 1 8 . B. 7. C. 14 . 4 D. 13. 1 . 4 11. 2 cos 2 sin 2 1 A. sin 2. 1 1 1 cos 1 cos A. 1. 2 B. . sin 2 C. . sin 2 1 D. . sin 2 B. cos 2. C. 2. 12. In the figure, ABC is a straight line. If DAB 90, DBA 50, DCA 25 and AD 10, BC D 10 50 A 3 where lies in quadrant III, 5 what is the value of tan cos ? 1 A. 20 31 B. 20 31 C. 20 1 D. 20 14. If sin D. 3 cos 2. 25 B C Chapter 7 Trigonometry 15. Which of the following statements is/are true for 180 x 270? A. B. C. D. I. sin x cos x 0 II. sin x tan x 0 III. cos x tan x 0 A. I and II only B. II only y D. II and III only 0 360 is 19 161 19 or 161 19, 161 or 270 19. What is the function represented by the following graph for 0 x 180? C. III only 16. The maximum value of 39 6 1 2 sin 2 1 for 4 A. 1. B. 0. 2 C. 1. 4 D. . 3 O 17. If sin cos 70, then A. 20 or 160. B. 70 or 110. C. 110 or 200. D. 120 or 240. 2 18. Solve the equation (3 sin 1)(sin 1) 0 for 0 x 360, correct your answers to the nearest degree if necessary. H INTS (for questions with x 30 A. y sin x B. y cos x C. y tan x 1 D. y tan x in the textbook) Revision Exercise 7 35. (c) Key information f () (2 sin 3) 2 6 Maximum value of f () for 0 360 31 Minimum value of f () for 0 360 7 60 90 120 150 180 40 New Trend Mathematics S4B — Supplement Analysis The range of may affect the maximum or minimum value of f (). Therefore, we have to analyze whether the angles corresponding to the maximum and minimum values for 0 360 lie in the range 0 180. Method Check the range of sin for 0 180 so as to find the range of f () for 0 180.