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anonymous

  • one year ago

Transform each polar equation to an equation in rectangular coordinates and identify its shape. a. θ = -(π / 6) b. r = (4 / (2cosθ - 3sinθ))

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  1. anonymous
    • one year ago
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    @dan815 @freckles @Luigi0210 @radar @rvc @sleepyjess

  2. anonymous
    • one year ago
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    I think for A, r = 2 but I'm not sure

  3. freckles
    • one year ago
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    \[\theta=\frac{-\pi}{6}\] actually describes a line |dw:1433651567694:dw|

  4. freckles
    • one year ago
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    so you know your line will be of the form y=mx

  5. freckles
    • one year ago
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    now the job is to find m

  6. anonymous
    • one year ago
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    Okay...

  7. freckles
    • one year ago
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    \[\text{ recall } \frac{y}{x}=\tan(\theta) \\\]

  8. freckles
    • one year ago
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    plug in your value of theta

  9. freckles
    • one year ago
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    you can also write that as y=tan(theta)*x

  10. anonymous
    • one year ago
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    I am so confused...

  11. anonymous
    • one year ago
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    I thought we were finding m? Are we going backwards now?

  12. freckles
    • one year ago
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    m is just a constant value I showed what m was by replacing m with tan(theta) and I got m was tan(theta) by using the equation y/x=tan(theta)

  13. anonymous
    • one year ago
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    ohhh, so how do I find out what x and y are since I already know theta?

  14. freckles
    • one year ago
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    I knew my equation would look like this because I knew theta=a number is a line \[y=mx\]

  15. freckles
    • one year ago
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    y and x are variables

  16. freckles
    • one year ago
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    they vary

  17. freckles
    • one year ago
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    you only need to find m aka tan(theta)

  18. anonymous
    • one year ago
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    But I already know what theta is for a. I need to know how to find r, right?

  19. anonymous
    • one year ago
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    @amilapsn

  20. anonymous
    • one year ago
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    @UsukiDoll

  21. amilapsn
    • one year ago
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    :)

  22. anonymous
    • one year ago
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    Can you help me?

  23. amilapsn
    • one year ago
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    You don't need to find r

  24. anonymous
    • one year ago
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    Then how am I supposed to graph it, if I don't know what r is?

  25. amilapsn
    • one year ago
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    You just need to find the Cartesian form of the polar equation \(\theta = -\frac{\pi}{6}\)

  26. anonymous
    • one year ago
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    Okay...how do I do that?

  27. anonymous
    • one year ago
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    Do I use tan(theta)

  28. anonymous
    • one year ago
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    or arctan (theta)?

  29. amilapsn
    • one year ago
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    You know that's a straight line, right?

  30. anonymous
    • one year ago
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    yes

  31. amilapsn
    • one year ago
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    Do you know how to get a equation of a line?

  32. anonymous
    • one year ago
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    no

  33. amilapsn
    • one year ago
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    you know the Cartesian form of the line is y=mx+c, right?

  34. anonymous
    • one year ago
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    okay, so how would I write that with -pi/6??

  35. anonymous
    • one year ago
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    hello...??? @amilapsn

  36. amilapsn
    • one year ago
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    Calm down... @Shorty1234 take a deep breath.... :)

  37. amilapsn
    • one year ago
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    yes of course.... You have to find m and c... what would be c, (it's easy)?

  38. anonymous
    • one year ago
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    -pi/6?

  39. amilapsn
    • one year ago
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    No... I'll ask again... you know the Cartesian form of the line is y=mx+c, right? What would m and c represent?

  40. amilapsn
    • one year ago
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    (Their meaning)

  41. anonymous
    • one year ago
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    m is the slop and c is the y-intercept

  42. amilapsn
    • one year ago
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    yes that's it what's the y intercept =c then?

  43. anonymous
    • one year ago
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    -pi/6

  44. amilapsn
    • one year ago
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    What do you mean by the y intercept?

  45. anonymous
    • one year ago
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    is that not what "c" represents ?

  46. amilapsn
    • one year ago
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    Yes, it is ....

  47. anonymous
    • one year ago
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    okay...so whats your point

  48. amilapsn
    • one year ago
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    y- intercept is the y coordinate where straight line cuts the y axis, right?

  49. anonymous
    • one year ago
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    yes

  50. freckles
    • one year ago
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    sorry lost internet connection do you remember the drawing I made (since you guys are talking about the y-intercept)|dw:1433653769876:dw| where does that cross the y-axis?

  51. amilapsn
    • one year ago
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    so?

  52. anonymous
    • one year ago
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    it doesnt...so is it zero?

  53. amilapsn
    • one year ago
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    :) @freckles

  54. amilapsn
    • one year ago
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    you got it..! :)

  55. freckles
    • one year ago
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    well it does at 0 (0 is a number :)) so that is how I got y=mx earlier

  56. amilapsn
    • one year ago
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    Now we need to find m right?

  57. anonymous
    • one year ago
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    so my answer for a would be (0, -pi/6)?

  58. amilapsn
    • one year ago
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    @Shorty1234

  59. freckles
    • one year ago
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    anyways I will let @amilapsn continue since I was gone so long

  60. amilapsn
    • one year ago
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    No....

  61. anonymous
    • one year ago
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    .....is there more stuff I have to do??

  62. amilapsn
    • one year ago
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    Yes you do...

  63. amilapsn
    • one year ago
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    you do have to find m, right?

  64. anonymous
    • one year ago
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    ok

  65. amilapsn
    • one year ago
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    How do you find m?

  66. amilapsn
    • one year ago
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    can you tell me the relation ship between - pi/6 and slope?

  67. anonymous
    • one year ago
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    y2 - y1 / x2 - x1

  68. amilapsn
    • one year ago
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    That's it! So m would be..............?

  69. anonymous
    • one year ago
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    0?

  70. freckles
    • one year ago
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    |dw:1433654449002:dw| I'm sorry I said I was going to stay away :p but can you replace -pi/6 and x and y using any trig ratio?

  71. freckles
    • one year ago
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    by the way that is a right triangle I made there

  72. amilapsn
    • one year ago
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    no.... Can you tell me the relationship between m and pi/6

  73. freckles
    • one year ago
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    relate *not replace

  74. anonymous
    • one year ago
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    I give up, this makes zero sense

  75. freckles
    • one year ago
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    do you know tan(theta)=opp/adj?

  76. anonymous
    • one year ago
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    yes

  77. freckles
    • one year ago
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    |dw:1433654806812:dw| so tan(-pi/6)=?

  78. amilapsn
    • one year ago
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    ......in terms of y and x?

  79. freckles
    • one year ago
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    yes using the right triangle above

  80. anonymous
    • one year ago
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    I got -0.577350269

  81. amilapsn
    • one year ago
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    can you express the same in terms of x and y ?

  82. anonymous
    • one year ago
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    I don't understand???

  83. amilapsn
    • one year ago
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    using the right triangle drawn by freckles...

  84. anonymous
    • one year ago
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    but i don't know what x is?

  85. amilapsn
    • one year ago
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    yes you don't indeed... use just x and y...

  86. anonymous
    • one year ago
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    so express x and y using x and y? that makes no sense

  87. freckles
    • one year ago
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    no express tan(-pi/6) in terms of x and y

  88. freckles
    • one year ago
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    using the right triangle

  89. freckles
    • one year ago
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    I have even labeled the opposite side with the correct measurement and the adjacent side too for the point (x,y)

  90. freckles
    • one year ago
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    |dw:1433655304381:dw|

  91. anonymous
    • one year ago
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    so opp/adj ??

  92. freckles
    • one year ago
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    where opp is y and adj is x right?

  93. freckles
    • one year ago
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    so can you say what tan(-pi/6)=? in terms of x and y now?

  94. anonymous
    • one year ago
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    tan (-pi/6) = y ???

  95. freckles
    • one year ago
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    almost

  96. freckles
    • one year ago
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    you forgot the x on denominator

  97. anonymous
    • one year ago
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    so tan(-pi/6) / x = y?

  98. freckles
    • one year ago
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    :( tan(theta)=y/x you know what theta is just plug in your theta

  99. freckles
    • one year ago
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    you are making this way complication your Cartesian equation is suppose to be in terms of y and x you know theta=-pi/6 so you will have an equation in terms of x and y if you replace the theta with -pi/6

  100. freckles
    • one year ago
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    lol i'm having trouble with english tonight :p

  101. anonymous
    • one year ago
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    .....this makes no sense

  102. freckles
    • one year ago
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    why ? I thought the visual made it pretty clear

  103. freckles
    • one year ago
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    |dw:1433655836599:dw| if this point is (x,y) then ...

  104. freckles
    • one year ago
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    |dw:1433655860761:dw|

  105. freckles
    • one year ago
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    tan(-pi/6)=opp/adj where opp=y and adj=x

  106. anonymous
    • one year ago
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    yeah, I dont get this...

  107. freckles
    • one year ago
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    let me just do an example I guess say we have \[\theta=\frac{\pi}{4}\] again this is a line going through the origin this is of the form y=mx we only need to determine m we can draw this line |dw:1433655974295:dw|

  108. freckles
    • one year ago
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    let's say point (x,y) is somewhere on there randomly|dw:1433656006691:dw| just choose a spot

  109. freckles
    • one year ago
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    |dw:1433656020961:dw|

  110. freckles
    • one year ago
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    well looking at this triangle I can make the observation that \[\tan(\frac{\pi}{4})=\frac{y}{x} \] I can be done here

  111. freckles
    • one year ago
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    or I can simplify

  112. freckles
    • one year ago
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    tan(pi/4)=1

  113. freckles
    • one year ago
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    \[1=\frac{y}{x} \\ \text{ I can also multiply } x \text{ on both sides } y=x\]

  114. freckles
    • one year ago
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    y=x is the Cartesian (aka rectangular form ) of the polar equation theta=pi/4

  115. freckles
    • one year ago
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    say we have theta=-pi/4 well that would just be tan(-pi/4)=y/x which is -1=y/x which is-x=y

  116. freckles
    • one year ago
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    say we have theta=pi/3 that would just be tan(pi/3)=y/x or y=tan(pi/3)x we can evaluate tan(pi/3) \[\tan(\frac{\pi}{3})=\frac{\sqrt{3}}{1}=\sqrt{3}\] so theta=pi/3 can be written as the Cartesian equation \[y=\sqrt{3}x\]

  117. anonymous
    • one year ago
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    So my answer would be y = tan(-pi/6)x ?

  118. freckles
    • one year ago
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    yes and simplify the tan(-pi/6)

  119. anonymous
    • one year ago
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    right,I'll do that. So what about the second problem. Hopefully it what take as long to figure out....

  120. freckles
    • one year ago
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    b. r = (4 / (2cosθ - 3sinθ)) multiply both sides by the denominator on right hand side you will need the other two equations to help you to convert this in terms of x and y that you will need rcos(theta)=x and rsin(theta)=y

  121. anonymous
    • one year ago
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    ok, thanks

  122. freckles
    • one year ago
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    by the way these equations are useful when converting from one to the other \[r^2=x^2+y^2 \\ \tan(\theta)=\frac{y}{x} \\ x=r \cos(\theta) \\ y=r \sin(\theta)\] these are your tools to work these particular puzzles

  123. freckles
    • one year ago
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    though it is kinda useful knowing why they are and that is why I drew the picture above because it doesn't appear you know why these equations are

  124. freckles
    • one year ago
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    |dw:1433657166077:dw| all of those equations can be found just by using this picture

  125. amilapsn
    • one year ago
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    |dw:1433657937284:dw|

  126. amilapsn
    • one year ago
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    Just a hint: )

  127. amilapsn
    • one year ago
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    Things may seem complicated... But they aren't..! Math is the way to sort things out!! :)

  128. amilapsn
    • one year ago
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    Hope is the only thing you should have!!

  129. amilapsn
    • one year ago
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    Don't give up.. Just try....!

  130. amilapsn
    • one year ago
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    You'll find the mathematician inside you..

  131. amilapsn
    • one year ago
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    |dw:1433658575750:dw|

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