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Name: ________________________ Class: ___________________ Date: __________ ID: A Test 3 Trig Functions Multiple Choice Identify the choice that best completes the statement or answers the question. ____ 1. What is the value of sec 70 to the nearest thousandth? A. 0.342 B. 0.364 ____ 2. What is the value of cos (295) to the nearest thousandth? A. 2.364 B. –0.423 ____ C. –230º D. 40º 6. Which of these angles is NOT coterminal with an angle of 240 in standard position? A. 120º B. –600º ____ C. 50º D. –130º 5. Which of these angles is coterminal with an angle of 230 in standard position? A. –130º B. 130º ____ C. 0.052 D. –19.231 4. What is the measure of the reference angle for an angle of 310 in standard position? A. 310º B. –50º ____ C. 0.423 D. 2.145 3. What is the value of cos 453 to the nearest thousandth? A. –19.081 B. –0.052 ____ C. –2.924 D. 2.924 C. –60º D. 480º 7. Which expression represents the measures of all the angles coterminal with 301 in standard position? A. 301 k180, k Z B. 59 k360, k Z C. 301 k360, k ò D. 301 k360, k Z 1 Name: ________________________ ____ 8. Given tan A. cot ID: A 8 , which statement is true for all possible values of ? 9 9 8 9 8 8 C. cot 9 D. cot cannot be determined B. ____ cot 9. What is the length of the arc that subtends a central angle of 150 in the unit circle? A. 5 units 6 C. B. 75 units D. 6 units 5 5 units 12 ____ 10. In the unit circle, the length of the arc that subtends a positive central angle of is 1 units. 3 What is the measure of in degrees? 1 540 D. 60 A. 120 B. C. 60 ____ 11. In a circle with radius 8 units, the length of the arc that subtends a positive central angle of is 20 units. What is the measure of in degrees? 9 A. 100 B. 50 C. 50 D. 400 ____ 12. What is 85 in radians? A. 85 radians B. 15 300 radians C. D. 17 radians 36 17 radians 36 ____ 13. What is 240 in radians? A. –240 radians B. 4 radians 3 C. 43 200 radians 4 D. radians 3 2 Name: ________________________ ID: A ____ 14. What is 695 in radians? A. B. 139 radians 36 125 100 radians C. 695 radians D. 139 radians 36 ____ 15. What is –6 radians in degrees? A. –344 B. –19 ÊÁ 4 ____ 16. What is the value of sec ÁÁÁÁ Ë 7 A. 1.19 B. 1.00 C. –1080 D. –2 ˆ˜ ˜˜ to the nearest hundredth? ˜˜ ¯ C. –0.22 D. –4.49 ____ 17. What is the value of csc (3.7) to the nearest hundredth? A. 1.89 B. –15.50 C. 1.24 D. 0.53 ____ 18. In a circle with radius 5 cm, what is the length of the arc that subtends a central angle of 3 radians? Give the answer to the nearest tenth. 4 A. 11.8 cm B. 0.5 cm C. 3.8 cm D. 2.4 cm 2 ____ 19. Which angle is NOT coterminal with an angle of radians in standard position? 5 A. B. 12 5 8 5 C. 0 D. 3 18 5 Name: ________________________ ID: A ____ 20. Which function below has this graph? A. y sin x B. y tan x C. y cos x D. None of the above ____ 21. Which function below has this graph? A. y sec x B. y csc x C. y cot x D. None of the above 4 Name: ________________________ ID: A ____ 22. Which set of functions describes these graphs? A. y cos x y 2 cos x y 4 cos x B. y cos x y cos 2x y cos 4x C. y cos x ÊÁ y cos ÁÁÁÁ x Ë ÊÁ y cos ÁÁÁÁ x Ë ____ 23. What is the amplitude of the function y 9 sin x ? A. 9 B. 18 C. 9 D. –9 ____ 24. What is the amplitude of this sinusoidal function? A. 3 B. 2 C. 6 D. –3 5 D. 2 4 ˆ˜ ˜˜ ˜˜ ¯ ˆ˜ ˜˜ ˜˜ ¯ y cos x y cos x 2 y cos x 4 Name: ________________________ ID: A ____ 25. What is the period of the function y tan 5x ? A. B. 5 5 C. D. 2 5 5 ____ 26. What is the period of this function? y f(x) A. B. 2 4 C. D. 2 2 ÁÊ 5 ˜ˆ˜˜ ____ 27. What is the phase shift of the function y cos ÁÁÁÁ x ? 6 ˜˜¯ Ë A. B. 5 6 17 6 C. D. 7 6 5 6 ____ 28. This graph is the image of y cos x after a phase shift. Which value below could represent the phase shift? A. B. 13 6 11 6 C. D. 6 6 6 Name: ________________________ ID: A ____ 29. Which function below describes this graph? A. B. y sin (x 3 ) y sin x 3 C. D. y sin x y 3sin x C. D. y sin x 2 y 2sin x ____ 30. Which function below describes this graph? A. B. y sin x y sin 2x ____ 31. Which number is NOT in the domain of y tan 3x ? 7 A. 6 B. 4 C. 3 1 D. 3 7 Name: ________________________ ID: A ____ 32. Identify the transformations that would be applied to the graph of y sin x to get the graph of ÊÁ 1 ˆ˜ y sin ÁÁÁÁ x ˜˜˜˜ 1 . Ë2 ¯ A. A vertical stretch by a factor of 2, and then a translation of 1 unit down 1 B. A horizontal compression by a factor of , and then a translation of 1 unit down 2 C. A horizontal stretch by a factor of 2, and then a translation of 1 unit down 1 D. A horizontal stretch by a factor of 1, and then a translation of units right 2 ÊÁ 2 ˆ˜˜˜ 9? ____ 33. What is the amplitude of the graph of y 8 sin 2 ÁÁÁ x ÁË 5 ˜˜¯ A. 16 B. 2 C. –1 D. 8 ÊÁ 5 ˆ˜˜˜ 6? ____ 34. What is the equation of the centre line of the graph of the function y 2 cos 5 ÁÁÁ x ÁË 6 ˜˜¯ A. y –6 B. y –4 y –12 5 D. y 6 C. ÊÁ 5 ˆ˜˜˜ 8? ____ 35. What is the period of the function y 2 sin 3 ÁÁÁ x ÁË 7 ˜˜¯ A. B. 2 3 7 5 C. D. 5 7 3 2 ÊÁ ˆ˜ ____ 36. What is the range of the function y 2 cos 3 ÁÁÁ x ˜˜˜˜ 5 ? ÁË 5¯ A. 7 y 3 B. 3 y 7 C. 2 y 8 D. 3 y 7 8 Name: ________________________ ID: A ____ 37. Which function below best describes this graph? A. y sin B. y sin 3 3 (x 2 ) 5 C. y sin (x 2 ) 5 D. y sin 3 3 (x 2 ) 5 (x 2 ) 5 2 (x 4.5 ) 23 models the height, y metres, of a seat on a Ferris wheel at 9 any time x minutes after the wheel begins to rotate. What is the diameter of the wheel? ____ 38. Suppose the function y 19cos A. 19 m B. 9 m C. 38 m D. 42 m ____ 39. Which of the following angles, in degrees, is coterminal with, but not equal to, A. 36° B. 216° C. 306° D. 396° ____ 40. Determine the equation of a circle with centre at the origin and radius 8. A. x 2 y 2 8 C. x 2 y 2 16 B. x 2 y 2 64 D. x 2 y 2 9 8 1 radians? 5 Name: ________________________ ID: A ____ 41. Which graph represents an angle in standard position with a measure of 285°? C. A. B. D. ____ 42. Determine the measure of the angle in standard position shown on the graph below. Round your answer to the nearest tenth of a degree. A. 291.8° B. 201.8° C. 111.8° D. 21.8° 10 Name: ________________________ ID: A ____ 43. The coordinates of the point that lies at the intersection of the terminal arm and the unit circle at an angle of 33° are A. (0.84, 0.65) C. (0.84, 0.54) B. (0.65, 0.54) D. (0.54, 0.84) ____ 44. Identify the point on the unit circle corresponding to an angle of 3 1 , ) 2 2 C. ( 3 3 , ) 2 3 3 1 ( , ) 2 2 D. ( 3 1 , ) 3 2 A. ( B. radians in standard position. 6 ____ 45. Which point on the unit circle corresponds to tan = 0? A. (1,1) C. (0,0) D. (1,0) B. (0,1) ____ 46. If the angle is 1400° in standard position, it can be described as having made 8 8 A. 3 rotations C. 3 rotations 9 9 7 7 B. 7 rotations D. 7 rotations 9 9 11 Name: ________________________ ID: A 5 ____ 47. Which graph represents the function y = 2cos( x), where x is in degrees? 3 A. C. B. D. ____ 48. The graph of y sin x can be obtained by translating the graph of y cos x A. units to the right C. units to the right 4 units to the right D. units to the right B. 2 3 ____ 49. Give an equation for a transformed sine function with an amplitude of to the right, and a vertical translation of 3 units down. 9 A. y = sin 4 (x 7 / 8 ) – 3 C. y = 7 9 È ˘ B. y = sin ÍÍÍÎ 4 (x 7 / 8 ) ˙˙˙˚ 3 D. y = 7 9 1 7 , a period of , a phase shift of rad 7 2 8 9 ÈÍ ˘ sin Í 4 (x 7 / 8 ) ˙˙˙˚ 3 7 ÍÎ 9 sin 4 (x 7 / 8 ) – 3 7 ____ 50. Which of the following is not an asymptote of the function f () tan ? 9 C. x = A. x 2 7 3 B. x = D. x = 2 2 12 Name: ________________________ ID: A Short Answer 1. Sketch the angle –40° in standard position, then identify the reference angle. 2. Determine the exact value of sin(120). 3. Determine the exact value of csc 405. 4. For the point P(2,4) on the terminal arm of an angle in standard position, determine the exact value of cot . 5. Sketch the angle 1 radians in standard position. 2 6. The graph of y sin x is shown below for 0 x 2. Extend the graph for x 2 and for x 0 . 13 Name: ________________________ ID: A 7. The graph of y cos x is shown below. On the same grid, sketch the graph of y 4 cos x . 8. The graph of y cos x is shown below. On the same grid, sketch the graph of y cos 2x . y cos x 9. The graph of y cos x is shown below. On the same grid, sketch the graph of y cos x 2 . y cos x 14 Name: ________________________ ID: A 10. The graph of y sin x is shown below. ÊÁ ˆ˜ On the same grid, sketch the graph of y sin ÁÁÁ x ˜˜˜˜ . ÁË 6¯ y sin x 11. Write a general equation for the asymptotes of the graph of y tan (4x) . ÊÁ ˆ˜ 12. Identify the following characteristics of the graph of y 3cos 2 ÁÁÁÁ x ˜˜˜˜ shown below. 2¯ Ë • amplitude • phase shift • minimum value • period • zeros • maximum value • equation of the centre line • domain • range ÁÊ ˜ˆ y 3cos 2 ÁÁÁÁ x ˜˜˜˜ 2¯ Ë 13. Write an equation for a sine function with amplitude 8, period shift 4 . 15 2 , equation of centre line y 9 , and phase 3 Name: ________________________ ID: A 14. Write an equation that represents the sine function graphed below. 15. A table fan has a mark on the tip of one blade. The equation y 17cos (6 x) 28 represents the height of the mark, y centimetres, above the table x seconds after the fan is turned on. What is the height of the mark above the table when it is closest to the table? ÈÍ Ê ˆ ˘˙ 2 ÈÍ Ê ˆ ˘˙ 2 ÍÍ Á 5 ˜˜ ˙˙ Í Á ˜˙ ˜ ˙˙ ÍÍÍÍ sin ÁÁÁ 5 ˜˜˜ ˙˙˙˙ . 16. Find the exact value of ÍÍÍ cos ÁÁÁ ÍÍ ÁË 6 ˜˜¯ ˙˙˙ ÍÍ ÁË 6 ˜¯ ˙˙ ˚ ˚ Î Î Problem 1. P(3,5) is a terminal point of angle in standard position. Determine all possible measures of in the domain 740 20. Give the answers to the nearest degree. 2. Given cot 1, determine all possible measures of angle in the domain 2 2 . 16 Name: ________________________ ID: A 3. Graph y 2sin x . Identify the amplitude, period, general expression for the zeros, domain of the function, and range of the function. 4. Graph y sin 4x . Identify the amplitude, period, general expression for the zeros, general equation for the asymptotes, domain of the function, and range of the function. 17 Name: ________________________ ID: A ÁÊ ˜ˆ 5. The graph of y sin x is shown below. On the same grid, sketch the graph of y 4 sin 2 ÁÁÁÁ x ˜˜˜˜ 1. 4¯ Ë Describe these characteristics of this function: amplitude, period, phase shift, equation of the centre line, domain, and range y sin x ÊÁ 2 ˆ˜˜˜ 6. Sketch the graph of y 4sin ÁÁÁ 4x 2. ÁË 3 ˜˜¯ Describe these characteristics of the function: amplitude, period, phase shift, equation of the centre line, domain, and range 18 ID: A Test 3 Trig Functions Answer Section MULTIPLE CHOICE 1. ANS: REF: LOC: 2. ANS: REF: LOC: 3. ANS: REF: LOC: 4. ANS: REF: LOC: KEY: 5. ANS: REF: LOC: KEY: 6. ANS: REF: LOC: KEY: 7. ANS: REF: LOC: 8. ANS: REF: LOC: 9. ANS: REF: TOP: 10. ANS: REF: TOP: KEY: 11. ANS: REF: TOP: KEY: 12. ANS: LOC: 13. ANS: LOC: D PTS: 1 DIF: Easy 6.1 Trigonometric Ratios for Any Angle in Standard Position 12.T3 TOP: Trigonometry KEY: Procedural Knowledge C PTS: 1 DIF: Easy 6.1 Trigonometric Ratios for Any Angle in Standard Position 12.T3 TOP: Trigonometry KEY: Procedural Knowledge B PTS: 1 DIF: Easy 6.1 Trigonometric Ratios for Any Angle in Standard Position 12.T3 TOP: Trigonometry KEY: Procedural Knowledge C PTS: 1 DIF: Easy 6.1 Trigonometric Ratios for Any Angle in Standard Position 12.T1 TOP: Trigonometry Conceptual Understanding | Procedural Knowledge A PTS: 1 DIF: Easy 6.1 Trigonometric Ratios for Any Angle in Standard Position 12.T1 TOP: Trigonometry Conceptual Understanding | Procedural Knowledge C PTS: 1 DIF: Easy 6.1 Trigonometric Ratios for Any Angle in Standard Position 12.T1 TOP: Trigonometry Conceptual Understanding | Procedural Knowledge D PTS: 1 DIF: Easy 6.1 Trigonometric Ratios for Any Angle in Standard Position 12.T1 TOP: Trigonometry KEY: Conceptual Understanding A PTS: 1 DIF: Easy 6.1 Trigonometric Ratios for Any Angle in Standard Position 12.T3 TOP: Trigonometry KEY: Conceptual Understanding A PTS: 1 DIF: Easy 6.2 Angles in Standard Position and Arc Length LOC: 12.T1 Trigonometry KEY: Procedural Knowledge | Conceptual Understanding D PTS: 1 DIF: Moderate 6.2 Angles in Standard Position and Arc Length LOC: 12.T1 Trigonometry Procedural Knowledge | Conceptual Understanding | Problem-Solving Skills C PTS: 1 DIF: Moderate 6.2 Angles in Standard Position and Arc Length LOC: 12.T1 Trigonometry Procedural Knowledge | Conceptual Understanding | Problem-Solving Skills D PTS: 1 DIF: Easy REF: 6.3 Radian Measure 12.T1 TOP: Trigonometry KEY: Procedural Knowledge D PTS: 1 DIF: Easy REF: 6.3 Radian Measure 12.T1 TOP: Trigonometry KEY: Procedural Knowledge 1 ID: A 14. ANS: LOC: 15. ANS: LOC: 16. ANS: LOC: 17. ANS: LOC: 18. ANS: LOC: KEY: 19. ANS: LOC: KEY: 20. ANS: REF: TOP: 21. ANS: REF: TOP: 22. ANS: LOC: 23. ANS: LOC: KEY: 24. ANS: LOC: KEY: 25. ANS: LOC: 26. ANS: LOC: 27. ANS: LOC: 28. ANS: LOC: KEY: 29. ANS: LOC: 30. ANS: LOC: 31. ANS: LOC: KEY: 32. ANS: REF: TOP: A PTS: 1 DIF: Easy REF: 6.3 Radian Measure 12.T1 TOP: Trigonometry KEY: Procedural Knowledge C PTS: 1 DIF: Easy REF: 6.3 Radian Measure 12.T1 TOP: Trigonometry KEY: Procedural Knowledge D PTS: 1 DIF: Easy REF: 6.3 Radian Measure 12.T3 TOP: Trigonometry KEY: Procedural Knowledge A PTS: 1 DIF: Easy REF: 6.3 Radian Measure 12.T3 TOP: Trigonometry KEY: Procedural Knowledge A PTS: 1 DIF: Easy REF: 6.3 Radian Measure 12.T1 TOP: Trigonometry Procedural Knowledge | Conceptual Understanding C PTS: 1 DIF: Moderate REF: 6.3 Radian Measure 12.T1 TOP: Trigonometry Conceptual Understanding | Procedural Knowledge B PTS: 1 DIF: Easy 6.4 Graphing Trigonometric Functions LOC: 12.T4 Trigonometry KEY: Conceptual Understanding A PTS: 1 DIF: Difficult 6.4 Graphing Trigonometric Functions LOC: 12.T4 Trigonometry KEY: Conceptual Understanding | Problem-Solving Skills A PTS: 1 DIF: Easy REF: 6.5 Trigonometric Functions 12.T4 TOP: Trigonometry KEY: Conceptual Understanding C PTS: 1 DIF: Easy REF: 6.5 Trigonometric Functions 12.T4 TOP: Trigonometry Procedural Knowledge | Conceptual Understanding A PTS: 1 DIF: Easy REF: 6.5 Trigonometric Functions 12.T4 TOP: Trigonometry Procedural Knowledge | Conceptual Understanding D PTS: 1 DIF: Easy REF: 6.5 Trigonometric Functions 12.T4 TOP: Trigonometry KEY: Procedural Knowledge A PTS: 1 DIF: Moderate REF: 6.5 Trigonometric Functions 12.T4 TOP: Trigonometry KEY: Procedural Knowledge A PTS: 1 DIF: Easy REF: 6.5 Trigonometric Functions 12.T4 TOP: Trigonometry KEY: Procedural Knowledge D PTS: 1 DIF: Moderate REF: 6.5 Trigonometric Functions 12.T4 TOP: Trigonometry Conceptual Understanding | Procedural Knowledge D PTS: 1 DIF: Easy REF: 6.5 Trigonometric Functions 12.T4 TOP: Trigonometry KEY: Conceptual Understanding B PTS: 1 DIF: Moderate REF: 6.5 Trigonometric Functions 12.T4 TOP: Trigonometry KEY: Conceptual Understanding A PTS: 1 DIF: Moderate REF: 6.5 Trigonometric Functions 12.T4 TOP: Trigonometry Conceptual Understanding | Procedural Knowledge C PTS: 1 DIF: Easy 6.6 Combining Transformations of Sinusoidal Functions LOC: 12.T4 Trigonometry KEY: Procedural Knowledge 2 ID: A 33. ANS: REF: TOP: 34. ANS: REF: TOP: 35. ANS: REF: TOP: 36. ANS: REF: TOP: 37. ANS: REF: TOP: 38. ANS: REF: TOP: 39. ANS: NAT: 40. ANS: NAT: 41. ANS: NAT: 42. ANS: NAT: 43. ANS: NAT: KEY: 44. ANS: NAT: NOT: 45. ANS: NAT: KEY: 46. ANS: NAT: NOT: 47. ANS: NAT: KEY: 48. ANS: NAT: KEY: 49. ANS: NAT: KEY: D PTS: 1 DIF: Easy 6.6 Combining Transformations of Sinusoidal Functions LOC: 12.T4 Trigonometry KEY: Procedural Knowledge A PTS: 1 DIF: Easy 6.6 Combining Transformations of Sinusoidal Functions LOC: 12.T4 Trigonometry KEY: Procedural Knowledge A PTS: 1 DIF: Moderate 6.6 Combining Transformations of Sinusoidal Functions LOC: 12.T4 Trigonometry KEY: Procedural Knowledge D PTS: 1 DIF: Moderate 6.6 Combining Transformations of Sinusoidal Functions LOC: 12.T4 Trigonometry KEY: Procedural Knowledge | Conceptual Understanding D PTS: 1 DIF: Moderate 6.7 Applications of Sinusoidal Functions LOC: 12.T4 Trigonometry KEY: Procedural Knowledge | Conceptual Understanding C PTS: 1 DIF: Moderate 6.7 Applications of Sinusoidal Functions LOC: 12.T4 Trigonometry KEY: Procedural Knowledge | Conceptual Understanding D PTS: 1 DIF: Average OBJ: Section 4.1 T1 TOP: Angles and Angle Measure KEY: radians | degrees B PTS: 1 DIF: Easy OBJ: Section 4.2 T2 TOP: Unit Circle KEY: unit circle | unit circle equation C PTS: 1 DIF: Easy OBJ: Section 4.1 T1 TOP: Angles and Angle Measure KEY: degrees D PTS: 1 DIF: Easy OBJ: Section 4.1 T1 TOP: Angles and Angle Measure KEY: radians C PTS: 1 DIF: Average OBJ: Section 4.3 T2 TOP: Trigonometric Ratios trigonometric ratios | unit circle | terminal arm | angle A PTS: 1 DIF: Average OBJ: Section 4.3 T2 TOP: Trigonometric Ratios KEY: exact value | unit circle | radians tan90 and tan270 do not include undefined D PTS: 1 DIF: Average OBJ: Section 4.3 T2 TOP: Trigonometric Ratios Unit Circle | exact value | tangent ratio C PTS: 1 DIF: Average OBJ: Section 4.1 T1 TOP: Angles and Angle Measure KEY: rotations | standard position Mixed numbers B PTS: 1 DIF: Difficult OBJ: Section 5.1 T4 TOP: Graphing Sine and Cosine Functions graph | amplitude | period | sinusoidal function D PTS: 1 DIF: Easy OBJ: Section 5.2 T4 TOP: Transformations of Sinusoidal Functions translation | primary trigonometric function A PTS: 1 DIF: Difficult OBJ: Section 5.2 T4 TOP: Transformations of Sinusoidal Functions transformations | equation | properties | sinusoidal function 3 ID: A 50. ANS: A NAT: T4 PTS: 1 DIF: Easy TOP: The Tangent Function KEY: OBJ: Section 5.3 asymptote | tangent function SHORT ANSWER 1. ANS: Reference angle: 40° PTS: REF: LOC: KEY: 2. ANS: 1 DIF: Easy 6.1 Trigonometric Ratios for Any Angle in Standard Position 12.T1 TOP: Trigonometry Procedural Knowledge | Conceptual Understanding | Communication sin (120) 3 2 PTS: 1 DIF: Moderate REF: 6.1 Trigonometric Ratios for Any Angle in Standard Position LOC: 12.T3 TOP: Trigonometry KEY: Procedural Knowledge | Conceptual Understanding 3. ANS: csc 405 2 PTS: REF: LOC: KEY: 4. ANS: 1 DIF: Moderate 6.1 Trigonometric Ratios for Any Angle in Standard Position 12.T3 TOP: Trigonometry Procedural Knowledge | Conceptual Understanding cot PTS: REF: LOC: KEY: 1 2 1 DIF: Moderate 6.1 Trigonometric Ratios for Any Angle in Standard Position 12.T3 TOP: Trigonometry Procedural Knowledge | Conceptual Understanding 4 ID: A 5. ANS: PTS: 1 DIF: Easy REF: 6.3 Radian Measure LOC: 12.T1 TOP: Trigonometry KEY: Procedural Knowledge | Conceptual Understanding | Communication 6. ANS: PTS: 1 LOC: 12.T4 7. ANS: DIF: Easy REF: 6.4 Graphing Trigonometric Functions TOP: Trigonometry KEY: Procedural Knowledge PTS: 1 DIF: Easy REF: 6.5 Trigonometric Functions LOC: 12.T4 TOP: Trigonometry KEY: Procedural Knowledge | Communication 5 ID: A 8. ANS: y cos 2x y cos x PTS: 1 DIF: Easy REF: 6.5 Trigonometric Functions LOC: 12.T4 TOP: Trigonometry KEY: Procedural Knowledge | Communication 9. ANS: y cos x y cos x 2 PTS: 1 DIF: Easy REF: 6.5 Trigonometric Functions LOC: 12.T4 TOP: Trigonometry KEY: Procedural Knowledge | Communication 6 ID: A 10. ANS: ÊÁ ˆ˜ y sin ÁÁÁ x ˜˜˜˜ ÁË 6¯ y sin x PTS: 1 DIF: Moderate REF: 6.5 Trigonometric Functions LOC: 12.T4 TOP: Trigonometry KEY: Procedural Knowledge | Communication 11. ANS: Equations may vary. For example: x (2k 1) 8 ,k Z PTS: 1 DIF: Moderate REF: 6.5 Trigonometric Functions LOC: 12.T4 TOP: Trigonometry KEY: Conceptual Understanding | Procedural Knowledge 12. ANS: • amplitude: 3 • period: • equation of the centre line: y 0 • • • • • • phase shift: 2 7 5 3 3 5 7 , , , , , , , zeros: 4 4 4 4 4 4 4 4 domain: 2 x 2 minimum value: –3 maximum value: 3 range: 3 y 3 PTS: 1 DIF: Easy REF: 6.6 Combining Transformations of Sinusoidal Functions LOC: 12.T4 TOP: Trigonometry KEY: Procedural Knowledge 13. ANS: ÊÁ ˆ˜ y 8 sin 3 ÁÁÁÁ x ˜˜˜˜ +9 4¯ Ë PTS: 1 DIF: Easy REF: 6.6 Combining Transformations of Sinusoidal Functions LOC: 12.T4 TOP: Trigonometry KEY: Conceptual Understanding | Procedural Knowledge 7 ID: A 14. ANS: Students’ answers may vary. For example: 2 (x 1 ) 2. y 2 sin 5 PTS: 1 DIF: Moderate REF: 6.7 Applications of Sinusoidal Functions LOC: 12.T4 TOP: Trigonometry KEY: Conceptual Understanding | Procedural Knowledge 15. ANS: 11 cm PTS: 1 DIF: Moderate REF: 6.7 Applications of Sinusoidal Functions LOC: 12.T4 TOP: Trigonometry KEY: Conceptual Understanding | Procedural Knowledge 16. ANS: ÈÍ Ê ˆ ˘˙ 2 ÈÍ Ê ˆ ˘˙ 2 ÊÁ ˆ˜ 2 Ê ˆ 2 ˜˜ ÍÍ ÁÁ 5 ˜˜ ˙˙ ÍÍ ÁÁ 5 ˜˜ ˙˙ ÁÁ 3 ÁÁ 1 ˜˜ ˜ ÍÍ cos ÁÁ ˜˜ ˙˙ ÍÍ sin ÁÁ ˜˜ ˙˙ ÁÁ ÍÍ Á 6 ˜ ˙˙ ÍÍ Á 6 ˜ ˙˙ ÁÁ 2 ˜˜˜ ÁÁÁ 2 ˜˜˜ ˜ ÍÎ Ë ¯ ˙˚ ÍÎ Ë ¯ ˙˚ Á Ë ¯ ¯ Ë 3 1 4 4 1 2 PTS: 1 DIF: Difficult TOP: Trigonometric Ratios OBJ: Section 4.3 NAT: T3 KEY: exact value | unit circle 8 ID: A PROBLEM 1. ANS: The terminal arm of angle lies in Quadrant 1. ÁÊ 5 ˜ˆ The reference angle is: tan 1 ÁÁÁÁ ˜˜˜˜ Ö 59 Ë3¯ In Quadrant 1, Ö 59 Sketch a diagram. The angles that are coterminal with 59 in the domain 740 20 are approximately: 59 360 301 301 360 661 Possible values of are approximately: 661 and 301. PTS: REF: LOC: KEY: 1 DIF: Moderate 6.1 Trigonometric Ratios for Any Angle in Standard Position 12.T3 TOP: Trigonometry Conceptual Understanding | Procedural Knowledge 9 ID: A 2. ANS: tan 1 Since tan is negative, the terminal arm of angle lies in Quadrant 2 or Quadrant 4. The reference angle is: tan 1 (1) 4 3 4 7 In Quadrant 4: 2 , or 4 4 In Quadrant 2: 4 , or Sketch a diagram. An angle that is coterminal with 5 3 2 4 4 An angle that is coterminal with 7 2 4 4 3 in the domain 2 2 is: 4 7 in the domain 2 2 is: 4 So, the possible measures of angle are 3 5 7 , , , and 4 4 4 4 PTS: 1 DIF: Moderate REF: 6.3 Radian Measure LOC: 12.T3 TOP: Trigonometry KEY: Procedural Knowledge | Conceptual Understanding 10 ID: A 3. ANS: The graph of y 2sin x is the image after the graph of y sin x has been stretched vertically by a factor of 2. The amplitude is 2. The period is 2. The zeros are k , k Z. The domain is x ò . The range is 2 y 2. PTS: 1 DIF: Moderate REF: 6.5 Trigonometric Functions LOC: 12.T4 TOP: Trigonometry KEY: Conceptual Understanding | Procedural Knowledge | Communication 11 ID: A 4. ANS: The graph of y sin 4x is the image after the graph of y sin x has been 1 horizontally compressed by a factor of . 4 y sin 4x The amplitude is 1. 1 The period is . 2 k , k Z. The zeros are 4 There are no asymptotes. The domain is x ò . The range is 1 y 1. PTS: 1 DIF: Moderate REF: 6.5 Trigonometric Functions LOC: 12.T4 TOP: Trigonometry KEY: Conceptual Understanding | Procedural Knowledge | Communication 12 ID: A 5. ANS: ÊÁ ˆ˜ 2 ÁÁÁ x ˜˜˜˜ 1 to y a sin b (x c ) d : ÁË 4¯ a 4, so the graph of y sin x is stretched vertically by a factor of 4, and the amplitude is 4. 1 2 b 2, so the graph is compressed horizontally by a factor of , and the period is , or . 2 2 Compare y 4 sin c , so the graph is translated units left, and the phase shift is . 4 4 4 d 1, so the graph is translated 1 unit up, and the centre line has equation: y 1 The domain is: x ò The maximum value of the function is: 1 4 5 The minimum value of the function is: 1 4 3 So, the range is: 3 y 5 ÊÁ ˆ˜ y 4 sin 2 ÁÁÁ x ˜˜˜˜ 1 ÁË 4¯ y sin x PTS: 1 DIF: Moderate REF: 6.6 Combining Transformations of Sinusoidal Functions LOC: 12.T4 TOP: Trigonometry KEY: Conceptual Understanding | Procedural Knowledge | Communication 13 ID: A 6. ANS: ÊÁ 2 Write y 4sin ÁÁÁ 4x ÁË 3 ÊÁ ˆ˜ y 4sin 4 ÁÁÁÁ x ˜˜˜˜ 2 6¯ Ë ˆ˜ ˜˜ 2 in the form y a sin b (x c ) d . ˜˜ ¯ Compare to y a sin b (x c ) d : a 4, b 4, c , and d 2 6 The graph of y sin x is stretched vertically by a factor of 4, compressed horizontally by a factor of reflected in the x-axis, and then translated 6 1 , 4 units left and 2 units up. ÁÊ 2 ˜ˆ˜˜ y 4sin ÁÁÁÁ 4x 2 3 ˜˜¯ Ë a 4, so the amplitude is 4. 2 b 4, so the period is , or . 4 2 c , so the phase shift is . 6 6 d 2, so the equation of the centre line is y 2. The domain is x ò . The maximum y-value is 4 units above the centre line and the minimum y-value is 4 units below the centre line, so the range is 2 y 6. PTS: 1 DIF: Difficult REF: 6.6 Combining Transformations of Sinusoidal Functions LOC: 12.T4 TOP: Trigonometry KEY: Conceptual Understanding | Procedural Knowledge | Communication 14