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JayDS

  • 3 years ago

Solve the following equations over [0°, 360°]. 1) 2 sin (x − 60)° = 1 2) √2 cos(x+90)°+1 = 0 Show working out thanks.

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  1. Callisto
    • 3 years ago
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    For question (1) 2 sin (x − 60)° = 1 Divide both sides by 2, what do you get?

  2. JayDS
    • 3 years ago
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    sin(x-60)°=1/2

  3. Callisto
    • 3 years ago
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    Yes. Next, take arcsine for both sides, what do you get?

  4. JayDS
    • 3 years ago
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    sorry, what's arcsine?

  5. JayDS
    • 3 years ago
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    opposite of sine?

  6. Callisto
    • 3 years ago
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    arcsine is the inverse of sine function.

  7. JayDS
    • 3 years ago
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    (x-60)°= sin-1 (1/2)

  8. JayDS
    • 3 years ago
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    sin^-1

  9. Callisto
    • 3 years ago
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    Yes. What does \(sin^{-1}\frac{1}{2}\) give you?

  10. JayDS
    • 3 years ago
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    30

  11. Callisto
    • 3 years ago
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    Yes. That's the angle in quadrant I, what about in quadrant IV?

  12. JayDS
    • 3 years ago
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    360-30=330 deg

  13. Chlorophyll
    • 3 years ago
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    Or it's the good habit to memorize 1/2 = sin 30

  14. JayDS
    • 3 years ago
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    I do remember it actually.

  15. Callisto
    • 3 years ago
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    Oh~ So.. you get x - 60 = 30 or x-60 = 330 Can you solve the two?

  16. JayDS
    • 3 years ago
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    x=90 or x=390

  17. Callisto
    • 3 years ago
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    Ah!!! My mistake!! The other angle should be in quadrant II... Since sine is positive in quadrant I and II.. I'm sorry!!

  18. Callisto
    • 3 years ago
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    I'm sorry... Just start it again! For question 1, 2 sin (x − 60)° = 1 Divide both sides by 2, which you've done just now. sin (x − 60)° = 1/2 Take arcsine for both sides, you get x-60 = 30 or x-60 = ______ (____is the angle in quadrant II, since sine of a angle in quadrant I and II gives you a positive value)

  19. JayDS
    • 3 years ago
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    kk

  20. Callisto
    • 3 years ago
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    Can you work out the ___ ?

  21. JayDS
    • 3 years ago
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    Would it be 90?

  22. Callisto
    • 3 years ago
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    Hmm.... We're still on x-60 = (angle in quadrant II), right?

  23. JayDS
    • 3 years ago
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    yeh but I'm a bit confused with your method.

  24. JayDS
    • 3 years ago
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    on that step.

  25. JayDS
    • 3 years ago
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    how I was taught to do it was to find the angle in the first quadrant first, in this case it is 90 I believe?

  26. JayDS
    • 3 years ago
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    and then using the angle that you found you use 180-that angle found= to get the answer

  27. Callisto
    • 3 years ago
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    Can we do it in this way? Solve siny = 1/2, what is/ are y?

  28. JayDS
    • 3 years ago
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    y= sin^-1 (1/2)

  29. Callisto
    • 3 years ago
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    which is equal to?

  30. JayDS
    • 3 years ago
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    30

  31. Callisto
    • 3 years ago
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    and?

  32. JayDS
    • 3 years ago
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    150?

  33. Callisto
    • 3 years ago
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    Right! So, you get y=30 or y=150, agree?

  34. JayDS
    • 3 years ago
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    yeh

  35. Callisto
    • 3 years ago
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    Now, back you your question. 2 sin (x − 60)° = 1 sin (x-60) = 1/2 So, in this case, let x-60 = y, you'll get siny = 1/2. and just now, you solve y, which is 30 or 150. So can you get x?

  36. JayDS
    • 3 years ago
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    x-60=30 or x-60=150 x=90 or x= 210

  37. Callisto
    • 3 years ago
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    Now, it's done~

  38. JayDS
    • 3 years ago
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    yeh but I'm still a bit confused with a part of it.

  39. Callisto
    • 3 years ago
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    Which part?

  40. JayDS
    • 3 years ago
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    near the last part?

  41. Callisto
    • 3 years ago
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    What is it exactly?

  42. JayDS
    • 3 years ago
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    "So, in this case, let x-60 = y, you'll get siny = 1/2. and just now, you solve y, which is 30 or 150. So can you get x?"

  43. JayDS
    • 3 years ago
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    wait, is it possible if u type out all the working out in logical order?

  44. Callisto
    • 3 years ago
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    Oh... perhaps I should write it in this way. 2sin(x-60) = 1 Let y = x-60, then the equation becomes 2siny = 1 So, siny = 1/2 And as what you did, y= 30 or y=150 So, replace y by x-60 x-60 = 30 or x-60 = 150 Solve x. It should be clearer now. Please ask if you still have questions. And sorry for my poor presentation just now :(

  45. Chlorophyll
    • 3 years ago
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    Or you can visualize the circle: 1/2 = sin30 = sin ( 180 - 30 )

  46. JayDS
    • 3 years ago
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    kk, I get the working out but yeh I may not remember how to do it your way as I was sort of taught a different way by my cousin and I absolutely hate these type of confusing type questions.

  47. Callisto
    • 3 years ago
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    It's okay. Just choose the best way for yourself. :) Can you do question 2?

  48. JayDS
    • 3 years ago
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    yep, well I'll try do it your way and I think I can only do half of the steps

  49. Callisto
    • 3 years ago
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    You can try to do it using your own way first :)

  50. Callisto
    • 3 years ago
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    I'll check it for you

  51. JayDS
    • 3 years ago
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    wait, so I've done this so far using your method, the rest I don't get. √2 cos(x+90)°+1 = 0 cos(x+90)° = -1/√2 let y = x + 90 therefore cos y = -1/√2 y = ............ or y = ............

  52. Callisto
    • 3 years ago
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    So far so good :) What are the values of y such that cosy = -1/√2

  53. JayDS
    • 3 years ago
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    not sure

  54. Chlorophyll
    • 3 years ago
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    Hint: -1/√2 = -√2/ 2

  55. Callisto
    • 3 years ago
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    Hmm.. Forget my method... Can you solve it using your OWN way??

  56. JayDS
    • 3 years ago
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    well it was my cousin's way and he only taught me just then. I still don't quite fully understand his method that's why I was planning to see if anyone could simplify it down for me on here.

  57. Callisto
    • 3 years ago
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    Can you do me a favour by showing how your cousin has taught you? You can use another question or simply this question if you like.

  58. JayDS
    • 3 years ago
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    tan (x+45°) = 1 x∈[0°,360°] x+45 = 45° (x+45°) ∈ [45°,405°] 180° + 45° or 405° x+45° = 45°, 225° or 405° x = 0°, 180° or 360°

  59. Chlorophyll
    • 3 years ago
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    Your cousin's way is exactly as same as our way!

  60. JayDS
    • 3 years ago
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    lol, really?

  61. Callisto
    • 3 years ago
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    I hope you don't mind me asking the following question. 1. Have you learnt CAST ? 2. Do you know something like cos(180 - x) = -cosx sin(180-x) = sinx and so on? 3. Are you familiar with the trigo functions of special angles (that are 0 , 30 , 45, 60, 90, and so on) ? It's practically the same

  62. JayDS
    • 3 years ago
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    yes, I have and I do know the trig functions that u mentioned

  63. Callisto
    • 3 years ago
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    Oh great. Sudden quiz. What is cos135?

  64. JayDS
    • 3 years ago
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    in exact value or degrees?

  65. Chlorophyll
    • 3 years ago
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    Hint: 135 < 180

  66. Callisto
    • 3 years ago
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    135 is in degrees. cos135 gives you an exact value.

  67. JayDS
    • 3 years ago
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    -cos45 = -Sqrt2/2

  68. Callisto
    • 3 years ago
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    Yes.

  69. Callisto
    • 3 years ago
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    Can we go back to your question 2 now?

  70. JayDS
    • 3 years ago
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    yes, please. It's late over here and I want to go to sleep lol.

  71. JayDS
    • 3 years ago
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    but I want to figure out how to do it properly first so I can do the other ones tomorrow.

  72. Callisto
    • 3 years ago
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    Don't worry, I have to do my quiz in an hour too :| Here's your work! √2 cos(x+90)°+1 = 0 cos(x+90)° = -1/√2 let y = x + 90 therefore cos y = -1/√2 Can you solve y now?

  73. JayDS
    • 3 years ago
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    wait. let me see... and good look for your quiz.

  74. JayDS
    • 3 years ago
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    well since the value is negative it has to be in the second or third quadrant?

  75. Callisto
    • 3 years ago
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    Yes. Another hint given by @Chlorophyll before : -1/√2 = - (1/√2) x (√2/√2) = - √2/2

  76. JayDS
    • 3 years ago
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    from there I'm a bit stuck as to how to solve for y.

  77. Chlorophyll
    • 3 years ago
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    Consider y as X!

  78. Callisto
    • 3 years ago
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    cosy = -√2/2 What is the value of angle such that cos(angle) = -√2/2? Another hint: refer to the sudden quiz

  79. JayDS
    • 3 years ago
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    yes, but how did you get the -√2/2?

  80. Callisto
    • 3 years ago
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    Ratinoalization?! -1/√2 = - (1/√2) x (√2/√2) = - √2/2

  81. JayDS
    • 3 years ago
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    can't u cancel the top and bottom Sqrt 2 and so u get -1/Sqrt2

  82. Callisto
    • 3 years ago
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    Oh yes, of course! But you need to know that cos45 = 1/√2

  83. Chlorophyll
    • 3 years ago
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    -√2/2 looks more familiar to you or not?

  84. JayDS
    • 3 years ago
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    yes, it is cos 45 right?

  85. JayDS
    • 3 years ago
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    -cos 45 in this case*

  86. Callisto
    • 3 years ago
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    -cos45 = cos(___ -45) = cos(____+45) ?

  87. JayDS
    • 3 years ago
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    -cos45 = cos(180-45) = cos (180+45)?

  88. Callisto
    • 3 years ago
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    Yes. So... cosy = -1/√2 y = ... or y= ...?

  89. JayDS
    • 3 years ago
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    wait... let me write something else down first.

  90. Callisto
    • 3 years ago
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    Take your time~

  91. Chlorophyll
    • 3 years ago
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    @JayDS @Callisto giant leap improvement here =)

  92. JayDS
    • 3 years ago
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    lol, no way, I haven't improved at all but Callisto has.

  93. Callisto
    • 3 years ago
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    What??!!! You haven't improved at all :'( ???

  94. JayDS
    • 3 years ago
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    √2 cos(x+90)°+1 = 0 cos(x+90)° = -1/√2 let y = x + 90 therefore cos y = -1/√2 cos y = - √2/2 hmm where to from here?

  95. JayDS
    • 3 years ago
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    I guess I have slightly lol, I'm just really stupid.

  96. Chlorophyll
    • 3 years ago
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    @JayDS Our goal here is to help YOU improved :)

  97. Callisto
    • 3 years ago
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    No one is stupid except me, you can't be stupid than me:| √2 cos(x+90)°+1 = 0 cos(x+90)° = -1/√2 let y = x + 90 So, the equation becomes cos y = -1/√2 Your duty: solve y. How...?!

  98. Callisto
    • 3 years ago
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    *delete ''you can't be stupid than me''

  99. JayDS
    • 3 years ago
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    haha thanks, I guess I'm just a logical person, like I things shown out clearly well in a way that I can understand of course and I'm slow at learning, well depends on what type of maths as well.

  100. JayDS
    • 3 years ago
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    I like*

  101. Chlorophyll
    • 3 years ago
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    @Callisto wanna me to take over so you have time with your quiz?

  102. JayDS
    • 3 years ago
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    sorry for taking up your time guys and thanks for your help so far.

  103. Callisto
    • 3 years ago
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    @Chlorophyll Ah... I forget my quiz :| I'll come back as soon as possible!!

  104. Chlorophyll
    • 3 years ago
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    @Callisto Good luck, Sis! Not to be worried I promise to a good job here :)

  105. Callisto
    • 3 years ago
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    @Chlorophyll Sis, I trust you :) Thanks too!!

  106. JayDS
    • 3 years ago
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    what's with the "Sis" lol

  107. Chlorophyll
    • 3 years ago
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    √2 cos(x+90)°+1 = 0 -> cos(x+90)° = -1 /√2 = -√2/2 Since cos value is negative, now replace ... into where -√2/2 is, can you!

  108. JayDS
    • 3 years ago
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    sorry, I'm confused about the replacing part.

  109. Chlorophyll
    • 3 years ago
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    Hint: the angle in the second and third quadrants

  110. JayDS
    • 3 years ago
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    yep, I know that but... I'm unsure how to use that

  111. JayDS
    • 3 years ago
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    second quadrant = pi - angle so it will be pi - sqrt2 /2 third quadrant = pi + angle so it will be pi + Sqrt 2/2 Is that correct?

  112. Chlorophyll
    • 3 years ago
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    cos(x+90)° = -√2/2 cos(x+90)° = -cos45 = cos(180-45) = cos (180+45)

  113. Chlorophyll
    • 3 years ago
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    So you have 2 cases here!

  114. Chlorophyll
    • 3 years ago
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    x+90° = 180-45 and .....

  115. JayDS
    • 3 years ago
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    yeh, that is the same as what I said earlier? pi - sqrt 2/2 and pi + sqrt 2/2 I believe.

  116. JayDS
    • 3 years ago
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    but then you would need to convert is back to degrees.

  117. Chlorophyll
    • 3 years ago
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    It's incorrect to connect the degree ( pi) with the radian ( arc length ) pi - sqrt 2/2

  118. JayDS
    • 3 years ago
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    kk, can you quickly explain this line "cos(x+90)° = -cos45 = cos(180-45) = cos (180+45)"

  119. JayDS
    • 3 years ago
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    oh wait, I think I got it.

  120. Chlorophyll
    • 3 years ago
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    Don't pay attention to the left part yet!

  121. Chlorophyll
    • 3 years ago
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    -√2/2 = -cos45 Now we wish to get rid of negative sign: -cos45 = cos ( 180 - 45) -cos45 = cos ( 180 + 45)

  122. Chlorophyll
    • 3 years ago
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    That's what Callisto has been working so hard to show you the concept!

  123. JayDS
    • 3 years ago
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    √2 cos(x+90)°+1 = 0 cos(x+90)° = -1/√2 let y = x + 90 So, the equation becomes cos y = -1/√2 cos y = -√2/2 cos y = -cos 45 cos (x+90) = -cos 45 cos45 = cos ( 180 - 45) cos45 = cos ( 180 + 45)

  124. Chlorophyll
    • 3 years ago
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    Actually now you've done with the right side, plug it into the equation: cos (x+90) = -cos 45 In the second quadrant: cos (x+90) = cos ( 180 - 45) -> x+90 = ....?

  125. JayDS
    • 3 years ago
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    x + 90 = 135???

  126. Chlorophyll
    • 3 years ago
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    -> x = ...?

  127. JayDS
    • 3 years ago
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    45

  128. Chlorophyll
    • 3 years ago
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    Similarly for the third quadrant case, can you do it yourself?

  129. JayDS
    • 3 years ago
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    well if x=45 wouldn't it just be 180+45=225?

  130. Chlorophyll
    • 3 years ago
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    cos (x+90) = cos ( 180 + 45) -> x+90 = ....?

  131. JayDS
    • 3 years ago
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    x + 90 = 225 x = 135

  132. Chlorophyll
    • 3 years ago
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    I suggest that you don't jump around, as we carefully show you step by step, just follow it one by one, and practice it yourself from now on!

  133. JayDS
    • 3 years ago
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    LOL, okay big sis ^^ I will practice tomorrow but it's getting late so I'm going to sleep now.

  134. Chlorophyll
    • 3 years ago
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    Okie, the solution already saved into your profile. Feel free to use it as your study note!

  135. JayDS
    • 3 years ago
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    really, how?

  136. JayDS
    • 3 years ago
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    is that a new function?

  137. Chlorophyll
    • 3 years ago
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    Click into your profile, you'll see all your questions and answers!

  138. Chlorophyll
    • 3 years ago
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    You can't see your own questions and answer in your profile?

  139. JayDS
    • 3 years ago
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    Maybe you can only do because you have already reached 99 smart points or something.

  140. JayDS
    • 3 years ago
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    SmartScore*

  141. Chlorophyll
    • 3 years ago
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    No, you can check anybody profiles!

  142. JayDS
    • 3 years ago
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    yeh kk, found it, it was on the left side, I didn't see it over there and I thought you couldn't click anything there.

  143. JayDS
    • 3 years ago
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    Well I g2g now Sis, thanks so much for the help and help me say thanks to Callisto as well.

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