D.E.V part 1

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D.E.V part 1

  1. 2. It’s been 3 years since the day Zack and Sally falling in love with each other.
  2. 3. Before that day, Zack, Sally’s boyfriend, has been thinking about this day long times before. He wants to give her a big surprise.
  3. 4. Before that day, Zack, Sally’s boyfriend, has been thinking about this day long times before. He wants to give her a big surprise.
  4. 5. Before that day, Zack, Sally’s boyfriend, has been thinking about this day long times before. He wants to give her a big surprise. What should I do to make her feel happy ?????
  5. 7. He’s thinking of ...
  6. 8. Romantic night at the 5 stars restaurant. He’s thinking of ... He’s thinking of ... He’s thinking of ...
  7. 9. Romantic night at the 5 stars restaurant. Or going shopping!!! He’s thinking of ...
  8. 10. Romantic night at the 5 stars restaurant. Or going shopping!!! Or a romantic movie He’s thinking of ...
  9. 11. Romantic night at the 5 stars restaurant. Or going shopping!!! Or a romantic movie Make their love symbols He’s thinking of ...
  10. 12. Romantic night at the 5 stars restaurant. Or going shopping!!! Or a romantic movie Make their love symbols Or walking together at night He’s thinking of ...
  11. 13. The answer is….
  12. 15. Hey! How about the Ferris Wheel??? That’s something new and really romantic….
  13. 16. Hey! How about the Ferris Wheel??? That’s something new and really romantic….
  14. 17. Hey! How about the Ferris Wheel??? That’s something new and really romantic….
  15. 18. Hey! How about the Ferris Wheel??? That’s something new and really romantic…. Hey that’s a great idea.
  16. 19. He starts planning everything for that special day. He starts planning everything for that special day.
  17. 20. That day is coming…..
  18. 21. They’re really happy spending time together….
  19. 22. Sally, you should try on that Ferris Wheel…
  20. 23. Sally, you should try on that Ferris Wheel… Where honey???
  21. 24. Do you see it??.. Whoa..!!..
  22. 25. She gets on the Ferris Wheel, Zack said he has something for her and she has to get on the wheel in other to see that.
  23. 26. <ul><li>Zack had planned to make a surprise for Sally. He will tell her to get on the Ferris wheel first, then Sally has to answer 2 questions in an definite time to see the special thing that Zack had prepared for only her…. </li></ul><ul><li>First he had to calculate the time he will give her to answer his 2 questions. </li></ul>
  24. 27. <ul><li>The Ferris Wheel has the maximum height is 32m and it stand 2m above from the round. It takes 6 minutes to go from the bottom of the wheel to the top of it. From the height of 28m, Sally will have the best view to see Zack’s present. </li></ul><ul><li>How long will it take to bring Sally to the best view point? </li></ul>
  25. 28. Solving:
  26. 29. Solving: First we have to draw the graph so that we can easily solve this problem. m min
  27. 30. Solving: First we have to draw the graph so that we can easily solve this problem. Maximum height : 32m 32 m min
  28. 31. Solving: First we have to draw the graph so that we can easily solve this problem. Maximum height : 32m 32 Stain 2m above the ground 2 m min
  29. 32. Solving: First we have to draw the graph so that we can easily solve this problem. Maximum height : 32m 32 Stain 2m above the ground 2 m min Go from bottom to the top : 6mins half period = 6mins 6
  30. 33. Solving: First we have to draw the graph so that we can easily solve this problem. Maximum height : 32m 32 Stain 2m above the ground 2 m min Go from bottom to the top : 6mins half period = 6mins 6 = 15 [ the distance from the max and min points to the Sinusoidal Axis.] 17 (An Amplitude) (32-2) 2
  31. 34. Solving: First we have to draw the graph so that we can easily solve this problem. Maximum height : 32m 32 Stain 2m above the ground 2 m min Go from bottom to the top : 6mins half period = 6mins 6 = 15 [ the distance from the max and min points to the Sinusoidal Axis.] 17 15 15 (An Amplitude) (32-2) 2
  32. 35. Solving: First we have to draw the graph so that we can easily solve this problem. Maximum height : 32m 32 Stain 2m above the ground 2 m min Go from bottom to the top : 6mins half period = 6mins 6 = 15 [ the distance from the max and min points to the Sinusoidal Axis.] 17 From the information above, we can sketch the graph. (An Amplitude) (32-2) 2
  33. 36. Solving: First we have to draw the graph so that we can easily solve this problem. Maximum height : 32m 32 Stain 2m above the ground 2 m min Go from bottom to the top : 6mins half period = 6mins 6 = 15 [ the distance from the max and min points to the Sinusoidal Axis.] 17 From the information above, we can sketch the graph. Sinusoidal axis (An Amplitude) (32-2) 2
  34. 37. Solving: First we have to draw the graph so that we can easily solve this problem. Maximum height : 32m 32 Stain 2m above the ground 2 m min Go from bottom to the top : 6mins half period = 6mins 6 = 15 [ the distance from the max and min points to the Sinusoidal Axis.] 17 From the information above, we can sketch the graph. Sinusoidal axis P = 12mins 12 (An Amplitude) (32-2) 2
  35. 38. Solving: First we have to draw the graph so that we can easily solve this problem. Maximum height : 32m 32 Stain 2m above the ground 2 m min Go from bottom to the top : 6mins half period = 6mins 6 = 15 [ the distance from the max and min points to the Sinusoidal Axis.] 17 From the information above, we can sketch the graph. Sinusoidal axis P = 12mins 12 Max point (An Amplitude) (32-2) 2
  36. 39. Solving: First we have to draw the graph so that we can easily solve this problem. Maximum height : 32m 32 Stain 2m above the ground 2 m min Go from bottom to the top : 6mins half period = 6mins 6 = 15 [ the distance from the max and min points to the Sinusoidal Axis.] 17 From the information above, we can sketch the graph. Sinusoidal axis P = 12mins 12 Max point Min point (An Amplitude) (32-2) 2
  37. 40. Sine Function: f(x) = A sin B (x – C ) + D 32 2 m min 6 17 12
  38. 41. Sine Function: f(x) = A sin B (x – C ) + D A = B = C = D = 32 2 m min 6 17 12
  39. 42. Sine Function: f(x) = A sin B (x – C ) + D A = B = C = D = 15 32 2 m min 6 17 12
  40. 43. Sine Function: f(x) = A sin B (x – C ) + D 32 2 m min 6 17 12 15 A = B = C = D = 2 π P = 2 π 12 = π 6
  41. 44. Sine Function: f(x) = A sin B (x – C ) + D A = B = C = D = 15 3 [the graph shifts to the right] = = 32 2 m min 6 17 12 2 π P 2 π 12 π 6
  42. 45. Sine Function: f(x) = A sin B (x – C ) + D A = B = C = D = 15 = = 3 [the graph shifts to the right] +17 32 2 m min 6 17 12 2 π P 2 π 12 π 6
  43. 46. Sine Function: f(x) = A sin B (x – C ) + D A = B = C = D = 15 = = +17 f(x) = A sin B (x – C ) + D A = B = C = D = = = 15 A = B = C = D = = = 3 [the graph shifts to the right] 32 2 m min 6 17 12 2 π P 2 π 12 π 6 2 π P 2 π 12 π 6 2 π P 2 π 12 π 6
  44. 47. Sine Function: f(x) = A sin B (x – C ) + D f(x) = A sin B (x – C ) + D 32 2 m min 6 17 12 15 A = B = C = D = 2 π P = 2 π 12 = π 6 3 [the graph shifts to the right] +17 π 6 15 sin +17 (x – 3 )
  45. 48. Sine Function: f(x) = A sin B (x – C ) + D f(x) = A sin B (x – C ) + D Sub the value f(x) = 28m we get: 3 [the graph shifts to the right] 32 2 m min 6 17 12 15 A = B = C = D = 2 π P = 2 π 12 = π 6 +17 π 6 15 sin +17 (x – 3 )
  46. 49. Sine Function: f(x) = A sin B (x – C ) + D f(x) = A sin B (x – C ) + D Sub the value f(x) = 28m we get: 28 = 32 2 m min 6 17 12 3 [the graph shifts to the right] A = B = C = D = 15 2 π P = 2 π 12 = π 6 +17 π 6 15 sin +17 (x – 3 ) π 6 15 sin +17 (x – 3 )
  47. 50. <ul><li>We know that: </li></ul>We know that: 28 = π 6 15 sin +17 (x – 3 )
  48. 51. <ul><li>We know that: </li></ul>We know that: sin π 6 6 6 = 6 6 6 6 6 6 1 2 28 = π 6 15 sin +17 (x – 3 )
  49. 52. <ul><li>We know that: </li></ul>We know that: sin π 6 6 6 = 6 6 6 6 6 6 1 2 sin π 6 6 6 = 6 6 6 6 6 6 1 2 sin π 6 6 6 = 6 6 6 6 6 6 1 2 28 = 15 x (x - 3 ) + 17 1 2 28 = π 6 15 sin +17 (x – 3 )
  50. 53. <ul><li>We know that: </li></ul>We know that: sin π 6 6 6 = 6 6 6 6 6 6 1 2 sin π 6 6 6 = 6 6 6 6 6 6 1 2 sin π 6 6 6 = 6 6 6 6 6 6 1 2 = 11 15 2 (x - 3 ) 28 = 15 x (x - 3 ) + 17 1 2 28 = π 6 15 sin +17 (x – 3 )
  51. 54. <ul><li>We know that: </li></ul>We know that: 11 = 15x - 45 2 sin π 6 6 6 = 6 6 6 6 6 6 1 2 sin π 6 6 6 = 6 6 6 6 6 6 1 2 sin π 6 6 6 = 6 6 6 6 6 6 1 2 28 = π 6 15 sin +17 (x – 3 ) = 11 15 2 (x - 3 ) 28 = 15 x (x - 3 ) + 17 1 2
  52. 55. <ul><li>We know that: </li></ul>We know that: 11 = 15x - 45 2 22 = 15x – 45 sin π 6 6 6 = 6 6 6 6 6 6 1 2 sin π 6 6 6 = 6 6 6 6 6 6 1 2 sin π 6 6 6 = 6 6 6 6 6 6 1 2 28 = π 6 15 sin +17 (x – 3 ) = 11 15 2 (x - 3 ) 28 = 15 x (x - 3 ) + 17 1 2
  53. 56. <ul><li>We know that: </li></ul>We know that: 11 = 15x - 45 2 22 = 15x – 45 67 = 15x sin π 6 6 6 = 6 6 6 6 6 6 1 2 sin π 6 6 6 = 6 6 6 6 6 6 1 2 sin π 6 6 6 = 6 6 6 6 6 6 1 2 28 = π 6 15 sin +17 (x – 3 ) = 11 15 2 (x - 3 ) 28 = 15 x (x - 3 ) + 17 1 2
  54. 57. <ul><li>We know that: </li></ul>We know that: 11 = 15x - 45 2 22 = 15x – 45 67 = 15x x = 4.47 mins sin π 6 6 6 = 6 6 6 6 6 6 1 2 sin π 6 6 6 = 6 6 6 6 6 6 1 2 sin π 6 6 6 = 6 6 6 6 6 6 1 2 28 = π 6 15 sin +17 (x – 3 ) = 11 15 2 (x - 3 ) 28 = 15 x (x - 3 ) + 17 1 2
  55. 58. <ul><li>We know that: </li></ul>We know that: 11 = 15x - 45 2 22 = 15x – 45 67 = 15x x = 4.47 mins ~ 4.5mins We could also solve this with cosine function. sin π 6 6 6 = 6 6 6 6 6 6 1 2 sin π 6 6 6 = 6 6 6 6 6 6 1 2 sin π 6 6 6 = 6 6 6 6 6 6 1 2 28 = π 6 15 sin +17 (x – 3 ) = 11 15 2 (x - 3 ) 28 = 15 x (x - 3 ) + 17 1 2
  56. 59. Therefore, Sally has about 4.5mins.
  57. 60. Sally has 3 boxes to choose that Zack had prepared before…
  58. 61. She picks the heart box first, and see a question.
  59. 62. If sinx = 7 12
  60. 63. If sinx = Cosy = 7 12 1 2
  61. 64. If sinx = Cosy = Find cos2x + Sin2y 7 12 1 2
  62. 65. If sinx = Cosy = Find cos2x + Sin2y From sinx, we could find cosx 7 12 1 2
  63. 66. If sinx = Cosy = Find cos2x + Sin2y From sinx, we could find cosx = 1 7 12 1 2 Sin x 2 + Cos x 2
  64. 67. If sinx = Cosy = Find cos2x + Sin2y From sinx, we could find cosx Sin x 2 + Cos x 2 - = 1 = 1 = 1 = 1 1 2 Cos x 2 Sin x 2 Sin x 2 Cos x 2 7 12
  65. 68. If sinx = Cosy = Find cos2x + Sin2y From sinx, we could find cosx Sin x 2 + Cos x 2 = 1 - = 1 = 1 = 1 - 2 = 1 1 2 Cos x 2 Sin x 2 Sin x 2 Cos x 2 Cos x 2 7 12 7 12 ( (
  66. 69. If sinx = Cosy = Find cos2x + Sin2y From sinx, we could find cosx Sin x 2 + Cos x 2 = 1 - = 1 = 1 = 1 - 2 = 1 - = 1 1 2 Cos x 2 Sin x 2 Sin x 2 Cos x 2 Cos x 2 7 12 7 12 ( ( Cos x 2 49 144
  67. 70. If sinx = Cosy = Find cos2x + Sin2y From sinx, we could find cosx Sin x 2 + Cos x 2 = 1 - = 1 = 1 = 1 - 2 = 1 - = = 1 1 2 Cos x 2 Sin x 2 Sin x 2 Cos x 2 Cos x 2 7 12 7 12 ( ( Cos x 2 49 144 Cos x 2 95 144
  68. 71. If sinx = Cosy = Find cos2x + Sin2y From sinx, we could find cosx Sin x 2 + Cos x 2 = 1 - = 1 = 1 = 1 - 2 = 1 - = Cosx = = 1 1 2 Cos x 2 Sin x 2 Sin x 2 Cos x 2 Cos x 2 7 12 7 12 ( ( Cos x 2 49 144 Cos x 2 95 144 √ 95 144
  69. 72. If sinx = Cosy = Find cos2x + Sin2y From sinx, we could find cosx Sin x 2 + Cos x 2 = 1 - = 1 = 1 = 1 - 2 = 1 - = Cosx = = 1 With the same method, we could find siny 1 2 Cos x 2 Sin x 2 Sin x 2 Cos x 2 Cos x 2 7 12 7 12 ( ( Cos x 2 49 144 Cos x 2 95 144 √ 95 144
  70. 73. If sinx = Cosy = Find cos2x + Sin2y From sinx, we could find cosx Sin x 2 + Cos x 2 = 1 - = 1 = 1 = 1 - 2 = 1 - = Cosx = = 1 With the same method, we could find siny Sin y 2 + 2 2 Cos y = 1 1 2 Cos x 2 Sin x 2 Sin x 2 Cos x 2 Cos x 2 7 12 7 12 ( ( Cos x 2 49 144 Cos x 2 95 144 √ 95 144 Sin y 2 2 2 Cos y
  71. 74. If sinx = Cosy = Find cos2x + Sin2y From sinx, we could find cosx Sin x 2 + Cos x 2 = 1 - = 1 = 1 = 1 - 2 = 1 - = Cosx = = 1 With the same method, we could find siny Sin y 2 + 2 2 Cos y = 1 = 1 - 1 2 Cos x 2 Sin x 2 Sin x 2 Cos x 2 Cos x 2 7 12 7 12 ( ( Cos x 2 49 144 Cos x 2 95 144 √ 95 144 Sin y 2 2 Sin y 2 2 2 Cos y 2 Cos y
  72. 75. If sinx = Cosy = Find cos2x + Sin2y From sinx, we could find cosx Sin x 2 + Cos x 2 = 1 - = 1 = 1 = 1 - 2 = 1 - = Cosx = = 1 With the same method, we could find siny Sin y 2 + 2 2 Cos y = 1 = 1 - = 1 - 1 2 Cos x 2 Sin x 2 Sin x 2 Cos x 2 Cos x 2 7 12 7 12 ( ( Cos x 2 49 144 Cos x 2 95 144 √ 95 144 Sin y 2 2 Sin y 2 2 2 Cos y 2 Cos y Sin y 2 2 1 2 ( ( 2
  73. 76. If sinx = Cosy = Find cos2x + Sin2y From sinx, we could find cosx Sin x 2 + Cos x 2 = 1 - = 1 = 1 = 1 - 2 = 1 - = Cosx = = 1 With the same method, we could find siny Sin y 2 + 2 2 Cos y = 1 = 1 - = 1 - = 1 - 1 2 Cos x 2 Sin x 2 Sin x 2 Cos x 2 Cos x 2 7 12 7 12 ( ( Cos x 2 49 144 Cos x 2 95 144 √ 95 144 Sin y 2 2 Sin y 2 2 2 Cos y 2 Cos y Sin y 2 2 1 2 ( ( 2 Sin y 2 2 1 4
  74. 77. If sinx = Cosy = Find cos2x + Sin2y From sinx, we could find cosx Sin x 2 + Cos x 2 = 1 - = 1 = 1 = 1 - 2 = 1 - = Cosx = = 1 With the same method, we could find siny Sin y 2 + 2 2 Cos y = 1 = 1 - = 1 - = 1 - = 1 2 Cos x 2 Sin x 2 Sin x 2 Cos x 2 Cos x 2 7 12 7 12 ( ( Cos x 2 49 144 Cos x 2 95 144 √ 95 144 Sin y 2 2 Sin y 2 2 2 Cos y 2 Cos y Sin y 2 2 1 2 ( ( 2 Sin y 2 2 1 4 Sin y 2 2 3 4
  75. 78. If sinx = Cosy = Find cos2x + Sin2y From sinx, we could find cosx Sin x 2 + Cos x 2 = 1 - = 1 = 1 = 1 - 2 = 1 - = Cosx = = 1 With the same method, we could find siny Sin y 2 + 2 2 Cos y = 1 = 1 - = 1 - = 1 - = Siny = 1 2 Cos x 2 Sin x 2 Sin x 2 Cos x 2 Cos x 2 7 12 7 12 ( ( Cos x 2 49 144 Cos x 2 95 144 √ 95 144 Sin y 2 2 Sin y 2 2 2 Cos y 2 Cos y Sin y 2 2 1 2 ( ( 2 Sin y 2 2 1 4 Sin y 2 2 3 4 √ 3 2 49 144
  76. 79. cos2x + sin2y = Cos2x = Cos(x + x) Sin2y = sin(y + y)
  77. 80. cos2x + sin2y = Cos2x = Cos(x + x) = cosxcosx - sinxsinx Sin2y = sin(y + y) = sinycosy + cosysiny
  78. 81. cos2x + sin2y = Cos2x = Cos(x + x) = cosxcosx - sinxsinx = cos x – sin x 2 2 Sin2y = sin(y + y) = sinycosy + cosysiny = 2sinycosy
  79. 82. cos2x + sin2y = Cos2x = Cos(x + x) = cosxcosx - sinxsinx = cos x – sin x 2 2 = - Sin2y = sin(y + y) = sinycosy + cosysiny = 2sinycosy 95 144 49 144 = 2 x 1 2 √ 3 2
  80. 83. cos2x + sin2y = Cos2x = Cos(x + x) = cosxcosx - sinxsinx = cos x – sin x 2 2 = - = Sin2y = sin(y + y) = sinycosy + cosysiny = 2sinycosy = 95 144 49 144 46 144 = 2 x 1 2 √ 3 2 = √ 3 2 √ 3 2 √ 3 2
  81. 84. cos2x + sin2y = Cos2x = Cos(x + x) = cosxcosx - sinxsinx = cos x – sin x 2 2 = - = Sin2y = sin(y + y) = sinycosy + cosysiny = 2sinycosy cos2x + sin2y = = 95 144 49 144 46 144 = 2 x 1 2 √ 3 2 = √ 3 2 √ 3 2 √ 3 2
  82. 85. cos2x + sin2y = Cos2x = Cos(x + x) = cosxcosx - sinxsinx = cos x – sin x 2 2 = - = Sin2y = sin(y + y) = sinycosy + cosysiny = 2sinycosy cos2x + sin2y = + = 95 144 49 144 46 144 = 2 x 1 2 √ 3 2 = √ 3 2 46 144 √ 3 2 √ 3 2 √ 3 2
  83. 86. cos2x + sin2y = Cos2x = Cos(x + x) = cosxcosx - sinxsinx = cos x – sin x 2 2 = - = Sin2y = sin(y + y) = sinycosy + cosysiny = 2sinycosy cos2x + sin2y = + = = + 95 144 49 144 46 144 = 2 x 1 2 √ 3 2 = √ 3 2 46 144 √ 3 2 √ 3 2 √ 3 2 46 144 72√3 144
  84. 87. cos2x + sin2y = Cos2x = Cos(x + x) = cosxcosx - sinxsinx = cos x – sin x 2 2 = - = Sin2y = sin(y + y) = sinycosy + cosysiny = 2sinycosy cos2x + sin2y = + = = + 46 + 72√3 = 144 95 144 49 144 46 144 = 2 x 1 2 √ 3 2 = √ 3 2 46 144 √ 3 2 √ 3 2 √ 3 2 46 144 72√3 144
  85. 88. cos2x + sin2y = Cos2x = Cos(x + x) = cosxcosx - sinxsinx = cos x – sin x 2 2 = - = Sin2y = sin(y + y) = sinycosy + cosysiny = 2sinycosy cos2x + sin2y = + = = + 46 + 72√3 = 144 = 95 144 49 144 46 144 = 2 x 1 2 √ 3 2 = √ 3 2 46 144 √ 3 2 √ 3 2 √ 3 2 46 144 72√3 144 2(23 + 36√3) 144
  86. 89. cos2x + sin2y = Cos2x = Cos(x + x) = cosxcosx - sinxsinx = cos x – sin x 2 2 = - = Sin2y = sin(y + y) = sinycosy + cosysiny = 2sinycosy cos2x + sin2y = + = = + 46 + 72√3 = 144 = = 95 144 49 144 46 144 = 2 x 1 2 √ 3 2 = √ 3 2 46 144 √ 3 2 √ 3 2 √ 3 2 46 144 72√3 144 2(23 + 36√3) 144 23 + 36√3 72
  87. 90. OMG!!! She has done with the first question, and starts to move on with the next one in the Noel box.
  88. 91. The next question is going to be harder than the first one…
  89. 92. The next question is going to be harder than the first one… cos2x 1 + tanx cotx 1 sin2x 2 - 1 = + sin x 2 -
  90. 93. How can I solve this one?? Uhm… let see…
  91. 94. cos2x 1 + tanx cotx 1 sin2x 2 - 1 = + sin x 2 - ● cos2x 1 + tanx cos2x 1 + tanx cos2x = = - 1 + 1 + tanx cos2x 1 + tanx cos2x cos x 2 sin x 2 1 + tanx cos2x 1 + tanx cos2x sinx cosx sin x 2
  92. 95. cos2x 1 + tanx cotx 1 sin2x 2 - 1 = + sin x 2 - ● cos2x 1 + tanx cos2x 1 + tanx cos2x = = - 1 + 1 + 1 + ☺ = cosx + sinx cosx cosx + sinx cosx cosx + sinx cosx cosx + sinx 1 + tanx cos2x 1 + tanx cos2x cos x 2 sin x 2 1 + tanx cos2x 1 + tanx cos2x sinx cosx sin x 2 sinx cosx sinx cosx sinx cosx 1 + sinx cosx 1 + cosx cosx + sinx
  93. 96. cos2x 1 + tanx cotx 1 sin2x 2 - 1 = + sin x 2 - ● cos2x 1 + tanx cos2x 1 + tanx cos2x = = - 1 + = - 1 + tanx cos2x 1 + tanx cos2x cos x 2 sin x 2 1 + tanx cos2x 1 + tanx cos2x sinx cosx sin x 2 cos x 2 sin x 2 cosx cosx + sinx
  94. 97. cos2x 1 + tanx cotx 1 sin2x 2 - 1 = + sin x 2 - ● cos2x 1 + tanx cos2x 1 + tanx cos2x = = - 1 + = - = - 1 x cosx cosx + sinx 1 + tanx cos2x 1 + tanx cos2x cos x 2 sin x 2 1 + tanx cos2x 1 + tanx cos2x sinx cosx sin x 2 cos x 2 sin x 2 cosx cosx + sinx sin x 2 cos x 2 sin x 2 - cos x 2 sin x 2 -
  95. 98. cos2x 1 + tanx cotx 1 sin2x 2 - 1 = + sin x 2 - ● cos2x 1 + tanx cos2x 1 + tanx cos2x = = - 1 + = - = - 1 x cosx cosx + sinx = cosx ( ) cosx + sinx 1 + tanx cos2x 1 + tanx cos2x cos x 2 sin x 2 1 + tanx cos2x 1 + tanx cos2x sinx cosx sin x 2 cos x 2 sin x 2 cosx cosx + sinx sin x 2 cos x 2 sin x 2 - cos x 2 sin x 2 - cos x 2 sin x 2 -
  96. 99. cos2x 1 + tanx cotx 1 sin2x 2 - 1 = + sin x 2 - ● cos2x 1 + tanx cos2x 1 + tanx cos2x = = - 1 + = - = - 1 x cosx cosx + sinx = cosx ( ) cosx + sinx = cosx (cosx + sinx) (cosx – sinx) cosx + sinx 1 + tanx cos2x 1 + tanx cos2x cos x 2 sin x 2 1 + tanx cos2x 1 + tanx cos2x sinx cosx sin x 2 cos x 2 sin x 2 cosx cosx + sinx sin x 2 cos x 2 sin x 2 - cos x 2 sin x 2 - cos x 2 sin x 2 -
  97. 100. cos2x 1 + tanx cotx 1 sin2x 2 - 1 = + sin x 2 - ● cos2x 1 + tanx cos2x 1 + tanx cos2x = = - 1 + = - = - 1 x cosx cosx + sinx = cosx ( ) cosx + sinx = cosx (cosx + sinx) (cosx – sinx) cosx + sinx 1 + tanx cos2x 1 + tanx cos2x cos x 2 sin x 2 1 + tanx cos2x 1 + tanx cos2x sinx cosx sin x 2 cos x 2 sin x 2 cosx cosx + sinx sin x 2 cos x 2 sin x 2 - cos x 2 sin x 2 - cos x 2 sin x 2 -
  98. 101. cos2x 1 + tanx cotx 1 sin2x 2 - 1 = + sin x 2 - ● cos2x 1 + tanx cos2x 1 + tanx cos2x = = - 1 + = - = - - 1 x cosx cosx + sinx = cosx ( ) cosx + sinx = cosx (cosx + sinx) (cosx – sinx) cosx + sinx 1 + tanx cos2x 1 + tanx cos2x cos x 2 sin x 2 1 + tanx cos2x 1 + tanx cos2x sinx cosx sin x 2 cos x 2 sin x 2 cosx cosx + sinx sin x 2 cos x 2 sin x 2 - cos x 2 sin x 2 cos x 2 sin x 2 - = cosx (cosx – sinx) ● cotx = cosx sinx
  99. 102. cos2x 1 + tanx cotx 1 sin2x 2 - 1 = + sin x 2 - ● cos2x 1 + tanx cos2x 1 + tanx cos2x = = - 1 + = - = - - 1 x cosx cosx + sinx = cosx ( ) cosx + sinx = cosx (cosx + sinx) (cosx – sinx) cosx + sinx = cosx (cosx – sinx) 1 + tanx cos2x 1 + tanx cos2x cos x 2 sin x 2 1 + tanx cos2x 1 + tanx cos2x sinx cosx sin x 2 cos x 2 sin x 2 cosx cosx + sinx sin x 2 cos x 2 sin x 2 - cos x 2 sin x 2 cos x 2 sin x 2 -
  100. 103. cos2x 1 + tanx cotx - 1 = cos2x 1 + tanx cos2x 1 + tanx cos2x = - 1 1 + tanx cos2x 1 sin2x 2 + sin x 2 - sin x 2 cosx sinx 1 sin2x 2 + sin x 2 - sin x 2 = cosx (cosx – sinx)
  101. 104. cos2x 1 + tanx cotx - 1 = cos2x 1 + tanx cos2x 1 + tanx cos2x = = - sinxcosx sinxcosx + sin x 2 - - sinxcosx 1 + tanx cos2x 1 sin2x 2 + sin x 2 - sin x 2 cosx sinx - 1 1 sin2x 2 + sin x 2 - sin x 2 = cosx (cosx – sinx) cosx sinx - 1 cos x 2 1 2 sin x 2 1 2
  102. 105. cos2x 1 + tanx cotx - 1 = cos2x 1 + tanx cos2x 1 + tanx cos2x = = - sinxcosx + sin x 2 - = + 1 + tanx cos2x 1 sin2x 2 + sin x 2 - sin x 2 cosx sinx - 1 1 sin2x 2 + sin x 2 - sin x 2 = cosx (cosx – sinx) cosx sinx - 1 cos x 2 sin x 2 sinxcosx 1 2 1 2 - sinxcosx cosx sinx - 1 cos x 2 sin x 2 - sinxcosx sinxcosx 1 2 1 2 - sinxcosx -
  103. 106. cos2x 1 + tanx cotx - 1 = cos2x 1 + tanx cos2x 1 + tanx cos2x = = - sinxcosx + sin x 2 - = + 1 + tanx cos2x 1 sin2x 2 + sin x 2 - sin x 2 cosx sinx - 1 1 sin2x 2 + sin x 2 - sin x 2 = cosx (cosx – sinx) cosx sinx - 1 cos x 2 sin x 2 sinxcosx 1 2 1 2 - sinxcosx cosx sinx - 1 cos x 2 sin x 2 - sinxcosx sinxcosx 1 2 1 2 - sinxcosx - cosx sinx - 1 = 1 - 2sinxcosx
  104. 107. cos2x 1 + tanx cotx - 1 = cos2x 1 + tanx cos2x 1 + tanx cos2x = = - sinxcosx + sin x 2 - = + = 1 - 2sinxcosx sin2x 1 + tanx cos2x 1 sin2x 2 + sin x 2 - sin x 2 cosx sinx - 1 1 sin2x 2 + sin x 2 - sin x 2 = cosx (cosx – sinx) cosx sinx - 1 cos x 2 sin x 2 sinxcosx 1 2 1 2 - sinxcosx cosx sinx - 1 cos x 2 sin x 2 - sinxcosx sinxcosx 1 2 1 2 - sinxcosx - cosx sinx - 1 = 1 - 2sinxcosx cosx sinx - 1 cosx sinx - 1 = 1 - cosx sinx - 1 = 1 -
  105. 108. cosx sinx - 1 = 1 - sin2x
  106. 109. = 1 - cosx sinx - sinx = cosx sinx - 1 1 - sin2x 1 - sin2x
  107. 110. = cosx sinx - sinx = cosx sinx - = sinx cosx sinx - 1 1 - sin2x 1 - sin2x (1 - sin2x)
  108. 111. = cosx sinx - sinx = sinx sinx ( cosx - sinx ) 2 cosx sinx - 1 1 - sin2x 1 - sin2x cosx sinx - = (1 - sin2x) cosx sinx - =
  109. 112. = cosx sinx - sinx = sinx - = 0 cosx sinx - 1 1 - sin2x 1 - sin2x cosx sinx - = (1 - sin2x) cosx sinx - = sinx ( cosx - sinx ) 2 cosx sinx - sinx ( cosx - sinx ) 2
  110. 113. = cosx sinx - sinx = sinx sinx ) 2 - = 0 [1 – sinx(cosx – sinx)] = 0 = 0 = [1 – sinx(cosx – sinx)] 0 = cosx sinx - 1 1 - sin2x 1 - sin2x cosx sinx - = (1 - sin2x) cosx sinx - = sinx ( cosx - sinx ) 2 cosx sinx - ( cosx - sinx ( cosx - sinx) ( cosx - sinx) [1 – sinx(cosx – sinx)] 0 =
  111. 114. [1 – sinx(cosx – sinx)] 0 = ( cosx - sinx)
  112. 115. [1 – sinx(cosx – sinx)] 0 = = = 0 0 [1 – sinx(cosx – sinx)] ( cosx - sinx) ( cosx - sinx)
  113. 116. [1 – sinx(cosx – sinx)] 0 = = = 0 0 [1 – sinx(cosx – sinx)] cosx = sinx 1 – sinx cosx + = 0 ( cosx - sinx) ( cosx - sinx) sin x 2
  114. 117. [1 – sinx(cosx – sinx)] 0 = = = 0 0 [1 – sinx(cosx – sinx)] cosx = sinx 1 – sinx cosx + = 0 tanx = 1 (b/c given tanx = 1) ( cosx - sinx) ( cosx - sinx) sin x 2
  115. 118. [1 – sinx(cosx – sinx)] 0 = = = 0 0 [1 – sinx(cosx – sinx)] cosx = sinx 1 – sinx cosx + = 0 tanx = 1 (b/c given tanx = 1) ( cosx - sinx) ( cosx - sinx) sin x 2 Divide both sides by cos x 2 X = π 4 5 π 4 X =
  116. 119. [1 – sinx(cosx – sinx)] 0 = = = 0 0 [1 – sinx(cosx – sinx)] cosx = sinx 1 – sinx cosx + = 0 tanx = 1 (b/c given tanx = 1) 1 – tanx + = 0 = 0 = ( cosx - sinx) ( cosx - sinx) sin x 2 cos x 2 tan x 2 X = π 4 5 π 4 X =
  117. 120. [1 – sinx(cosx – sinx)] 0 = = = 0 0 [1 – sinx(cosx – sinx)] cosx = sinx 1 – sinx cosx + = 0 tanx = 1 (b/c given tanx = 1) 1 = 0 = + 1 ( cosx - sinx) ( cosx - sinx) sin x 2 cos x 2 – tanx + tan x 2 0 = tan x 2 tan x 2 – tanx + tan x 2 0 = tan x 2 X = π 4 5 π 4 X =
  118. 121. [1 – sinx(cosx – sinx)] 0 = = = 0 0 [1 – sinx(cosx – sinx)] cosx = sinx 1 – sinx cosx + = 0 tanx = 1 (b/c given tanx = 1) 1 = 0 = + 1 – tanx + 0 = 2tan x 2 – tanx + 1 0 = ( cosx - sinx) ( cosx - sinx) sin x 2 cos x 2 – tanx + tan x 2 0 = tan x 2 tan x 2 tan x 2 tan x 2 X = π 4 5 π 4 X =
  119. 122. [1 – sinx(cosx – sinx)] 0 = = = 0 0 [1 – sinx(cosx – sinx)] cosx = sinx 1 – sinx cosx + = 0 tanx = 1 (b/c given tanx = 1) 1 = 0 = + 1 – tanx + 0 = 2tan x 2 – tanx + 1 0 = = (-1) 2 - 4(2)(1) ( cosx - sinx) ( cosx - sinx) sin x 2 cos x 2 – tanx + tan x 2 0 = tan x 2 tan x 2 tan x 2 tan x 2 X = π 4 5 π 4 X =
  120. 123. [1 – sinx(cosx – sinx)] 0 = = = 0 0 [1 – sinx(cosx – sinx)] cosx = sinx 1 – sinx cosx + = 0 tanx = 1 (b/c given tanx = 1) 1 = 0 = + 1 – tanx + 0 = 2tan x 2 – tanx + 1 0 = = (-1) 2 - 4(2)(1) = 1 – 8 = -7 ( cosx - sinx) ( cosx - sinx) sin x 2 cos x 2 – tanx + tan x 2 0 = tan x 2 tan x 2 tan x 2 tan x 2 X = π 4 5 π 4 X =
  121. 124. [1 – sinx(cosx – sinx)] 0 = = = 0 0 [1 – sinx(cosx – sinx)] cosx = sinx 1 – sinx cosx + = 0 tanx = 1 (b/c given tanx = 1) 1 = 0 = + 1 – tanx + 0 = 2tan x 2 – tanx + 1 0 = = (-1) 2 - 4(2)(1) = 1 – 8 = -7 < 0 [ O ] ( cosx - sinx) ( cosx - sinx) sin x 2 cos x 2 – tanx + tan x 2 0 = tan x 2 tan x 2 tan x 2 tan x 2 X = π 4 5 π 4 X =
  122. 125. X = π 4 5 π 4 X =
  123. 126. k π (K ͼ I) X = π 4 5 π 4 X = ● ● ● X = 5 π 4 _ + ● ● ● X = π 4 _ + k π (K ͼ I)
  124. 127. 4 mins 28 secs ….
  125. 128. Will You Marry Me ??? Made by TN – HN

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