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Achievement Standard 90637 Trigonometry 3.3 Achievement Questions 1. Give exact values for a) cosec 90° b) cot c) cos 6 4 2. Write the following as trigonometric ratios of a positive acute angle a) sin 150° b) cos 240° c) tan 120° d) 3. What are the exact values of a) cos 300° c) tan b) sin 315° 12 4. a) What is the value of tan 1 ? 5 b) What is the value of sec ? 4 3 5 12 5. A regular pentagon is inscribed in a circle of radius 15cm. What is the length of each side of the pentagon? C 25 6. a) What is the value of sin 2A ? b) What is the value of cos 2A ? A 7 B 24 7. If sin 1 x 0.911 radians, what is the value of x ? 8. cos 90 A is equal to a) cos A b) sin A c) -sin A d) sin 90 A c) -cot A d) tan 90 A 9. tan 180 A is equal to a) tan A Trigonometry 2009 b) cot A 1 90637 10. Express each example as a trigonometric ratio with an acute angle in radians 3 2 7 a) tan b) sin c) cos 4 3 4 In each of the following five problems, solve for a value of angle x in degrees (3 s.f.) 11. cos x 0.548 12. 2sin x 0.6 13. 3tan 2x 7.2 14. 4cos3x 2.5 15. 3cos 2 x 25 2 In the next five problems, solve for a value of x in radians (3 s.f.) 16. 3tan 2x 5 17. 2 sin 2x 1 18. cos 2 x 0.7 0.6 3 tan 2 x 1 2 1 20. sin 3 x 0.5 2 19. 21. Sketch the graph of y tan x for the interval 22. Sketch the graph of y cos 1 2 x for the interval 0 x 4 3– 2 1 – 3 1 2 3 1 2 3 1 2 3 2 23. Sketch the graph of y 2sin 2 x for the interval x 22 24. Sketch the graph of y 2 cos 2 x 1 for the interval x 2 25. From the sketch of the curve y A sin B x C D , find the values of A, B, C, D y 3 2 1 – 1 2 3 2 2 x – 2 – 3 Trigonometry 2009 2 90637 1– 3 1 2 3 2 – 22 2 26. From the sketch of the curve y A cos B x C D , find the values of A, B, C, D y 1 – 6– 4 1 2 3 4 5 1 2 3 3 2 – – 22 2 2 3 2 2 – 1 2 x – 2 – 3 27. From the sketch of the curve y A tan Bx D , find the values of A, B, C, D y 6 5 4 3 2 1 – – 2 – 1 2 3 2 2 x – 2 – 3 – 4 28. The height of sea water about the mean sea level due to tides is given by h A cos t , where t = time in hours after high tide and A is the height of high tide 4 in metres. High tide of 2.5m occurs at 2pm. i. Sketch the graph of h A cos t 4 ii. Find out the time when the first low tide occurs after 2pm. iii. Calculate when the height of the tide is first at 1m. Give your answer to the nearest minute. Trigonometry 2009 3 90637 29. A ferris wheel at a showground rotates clockwise about its centre A. Passengers get on at point B. A seat starts at point B and t seconds later it is h metres above ground level. h 8cos t 9 20 i. ii. iii. iv. v. A B How long does one complete rotation take? What is the length of the radius of the wheel? How high is B above the ground? How high above ground level does the seat rise? What is the height of the chair after 30 seconds? 30. A spring oscillates about a mid position. The length of the spring in mm after t seconds is given by l 128 24sin 4t i. ii. iii. iv. v. How long is the spring when it is stationary? What is the maximum length of the spring? What is the length of the spring after 3 4 of a full cycle? What is the period of the motion? What is the frequency of the motion? Merit Questions 3 5 31. If sin A and sin B , where A and B are acute angles, find the value of 5 13 a) sin A B b) cos A B (do not use a calculator) 32. Prove each of the following 1 a) 2sin 75 sin15 2 33. Sketch the graph of y cot x, 3 b) 2 cos sin 1 4 4 x 2 2 34. Sketch the graph of y cosecx, 0 x 2 3 3 35. Given tan A , tan B , find the value of tan A B 4 5 36. Given that cos A B cos A cos B sin A sin B , establish that cos 2 A 1 2sin 2 A Trigonometry 2009 4 90637 37. What is the value of sin 75 sin15 (without a calculator) cos 75 cos15 38. Express as a product of two trigonometric functions a) sin5 A sin3A b) cos A cos5 A 39. Express as a sum or difference of two trigonometric functions a) 2cos x 2 y sin 3x 4 y b) 2cos60 sin30 Prove the following identities 1 40. tan cot sin cos 41. 1 tan 2 1 2sin 2 1 tan 2 42. cos 2 A cos A sin A cos A sin A 43. sin 2 A tan A 1 cos 2 A 44. sin 2 x 45. 46. sin 2 x cos 2 x sec x sin x cos x sin 3 A 2 B sin A cos A cos 3 A 2 B 47. tan x 48. 2 tan x 1 tan 2 x cot A B 1 cos 2 x 1 cos 2 x cosec2 1 1 sec2 cosec2 2 2 cos 1 sin 49. If u 1 sin 1 1 sin , prove that cos u cos Solve the following equations, for the range of angles indicated. Trigonometry 2009 5 90637 50. cos 2 cos 0, in radians, 0 2 51. cos cos 3 sin sin 3 0, in radians, 0 2 52. sin 2 x cos x 1, in radians, 0 x 2 53. cos2 2 x 3sin 2 x 3 0, in radians, - x 54. 2sin 2 1, 0 360 55. Find the general solutions for sin3x sin x 0 (in radians) 56. A concrete boat ramp is at 30º to the horizontal. Jules marks the concrete at the highest point the water reaches and his friend Verne marks the lowest level reached by the water. The distance between the two marks is 3.2 metres. The two budding scientists then marked the halfway point between the high and low tide marks. They discovered that the movement of the tides was uniformly cyclic and occurred at 12 hour intervals. a) Assuming we start our clock when the water is at its lowest level, write a trig equation that will model the vertical movement of the tides. b) For what fraction of the cycle time will the water be above a mark on the concrete exactly 1 metre below the high tide mark? c) Where will the water level on the ramp be 10 hours after the start? Excellence Questions 57. Find the value of x given that 2sin 1 x 58. Given that cos 2 A 1 sin 2 A , deduce that cos 4 A 1 8sin 2 A cos 2 A 59. Write cos3A cos A 2 A and use the expansion of cos A B to derive the formula cos 3 A 4 cos3 A 3cos A 5 60. Find the exact value of sec 6 Trigonometry 2009 6 90637 61. Find the smallest positive value in degrees of tan 1 1.5 62. Prove the formula sin 2 cos 2 1 63. Write tan 3A as tan A 2 A and use the expansion to derive the formula for tan3A in terms of tan A 64. Writing tan 4 A as tan 2 A 2 A , establish the formula for tan 4A in terms of tan A 65. From the patterns established in the previous problems write the expansion of tan5A in terms of tan A 66. Establish the formula for tan A B C . If A, B, C are the three angles of a triangle, show that tan A tan B tan C tan A tan B tan C 67. A triangle has adjacent sides of 5cm and 7cm with an included angle of 70°. What is the length of the third side? 68. A triangle has sides of 5cm, 6cm and 7cm. Calculate the size of the second largest angle. If you can’t manage that, find the size of the second smallest angle. 69. Use the sine rule to find . Use the value of to find the area of the triangle. 5m th 46 7m 70. The cuboid has dimensions 12cm 8cm 5cm What angle does the diagonal AB make with the horizontal? B 8 A 5 12 71. Find the lengths of x and y. 1 3 2 x Trigonometry 2009 7 y 90637 T 72. The diagram to establish the formula for sin A B 1 unit R S P U Q Prove the following identities 1 cos 1 cos 73. cot cosec 74. 1 1 tan 2 A c osecA cos A sin A cos A sin A 75. sin 2 cos 2 1 cot sin 2 cos 2 1 76. sin 4 sin 2 tan 2 cos 4 cos 2 1 2 77. If sin x cos x , find tan x in terms of and . 78. If tan 2 2 tan 2 1, show that 2cos2 cos2 0 sin 4 x cos 4 x 1 for all values of x for which the expression sin 4 x cos 4 x 1 on the right hand side is defined. For what values of x in the interval 0 x 2 is the right hand side of the expression not defined? 79. Prove that tan 2 x Solve the following equations. 80. sin 3x cos x 0 Give the general solution in radians 81. 3cot 2 5 7cosec Give the general solution in radians Trigonometry 2009 8 90637 82. sin 2 x 2sin x cos x 8cos 2 x 0 Give the general solution in radians 83. sin 2 2 sin 2 1 0 , in radians, 0 84. sin 4x sin 2x 0 , in degrees, 180 x 180 85. cos3x cos 2x cos x 0 , in degrees 0 x 360 86. cos 2 2 x 3sin 2 x 3 0 , in radians 0 2 87. 2 tan 3 tan . Exact values of in the interval 0 2 1 tan 2 88. cos 2 x 1 Exact values of x in the interval 0 2 2 89. A balloon is vertically over a point which lies in a direct line between two observers a distance of 800m apart. The observers note respectively the angles of elevation of the balloon to be 59° and 34°. Find the height of the balloon. 90. Show that x Hh H h H h x d2 91. d1 A man stands on top of a wall of height h metres and observes the elevation of a wireless mast to be α. He then descends from the wall and finds the elevation to be β. Show that the height of the mast exceeds that of the man by h sin cos metres. sin h 92. h is the height of a vertical tower standing at a point C on a horizontal plane ABC tan sin Show that h sin A C d B Trigonometry 2009 9 90637 93. The angle of elevation of a tower CD from A is 15° and from B it is 19°. A and B are 60m apart. Find the distance x. D h 94. Show that h 19 15 60m A B d sin sin sin C x D h A d B C PROBLEMS Q2 (d) missing ratio Q21 missing interval Do you want grids on the graphs? Q28 ,29 I have changed to way the variable is written Is there too much excellence? Hardly anyone does it. Is Q56 merit or excellence? Q60 do you need “exact value” Q69 is angle 46 or 42. There is a line of writing above Q66 Do I include it?? Are basic sine and cosine rule questions really excellence????? Trigonometry 2009 10 90637 Trigonometry 2009 11 90637