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anonymous

  • 5 years ago

1. Algebraically transform r = -8sinθ into a Cartesian equation of a circle (in standard form).

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  1. amistre64
    • 5 years ago
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    algebraically eh.....

  2. amistre64
    • 5 years ago
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    r = -8sin(t) is a straight line right?

  3. anonymous
    • 5 years ago
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    no, it's a circle

  4. amistre64
    • 5 years ago
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    I see it :)...polars still get me a little lol

  5. anonymous
    • 5 years ago
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    Ha same here

  6. amistre64
    • 5 years ago
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    lets do I table to see where it going ...can we do that? t = 0,45,90,360....

  7. amistre64
    • 5 years ago
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    when its at 90; or straight up, we get r = -8 so this is a circle about the y axis with a radius of 8 centerd 4 below the origin...

  8. amistre64
    • 5 years ago
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    but I am prolly messing that up still....

  9. amistre64
    • 5 years ago
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    nah..im right lol

  10. amistre64
    • 5 years ago
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  11. anonymous
    • 5 years ago
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    well in my textbook theres a question similar to this, but i dont know how to get the last equation. r=6sinθ r²=6rsinθ x²y²=6y x²-6y+9+x²=9 (y-3)²+x²=3²

  12. amistre64
    • 5 years ago
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    yeah, I was gettin to that :) just wanted to make sure I had an answer key to go by ;)

  13. anonymous
    • 5 years ago
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    OOHHH sorry

  14. amistre64
    • 5 years ago
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    our equation is going to end up as: x^2 + (y+4)^2 = 16 r = -8sin(t)

  15. amistre64
    • 5 years ago
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    r = -8sin(t) ; muliply by r r^2 = -8rsin(t) ; convert to cartesians x^2 + y^2 = -8y ; +8y x^2 + y^2 +8 = 0 ; complete square for "y" x^2 + y^2 +8y +16 = 16 ; convert to circle equation x^2 + (y+4)^2 = 16 ; tada!!!

  16. amistre64
    • 5 years ago
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    you see thats a typo ...8y*

  17. anonymous
    • 5 years ago
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    At which step? and where did the 16 come from?

  18. amistre64
    • 5 years ago
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    by now you should be familiar with a process called "completing the square". remember it?

  19. amistre64
    • 5 years ago
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    step 4 I put in 8 instead of 8y ;)

  20. anonymous
    • 5 years ago
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    Oh ok I see the typo

  21. amistre64
    • 5 years ago
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    a complete square can be transformed back and forth between to ...... forms.

  22. amistre64
    • 5 years ago
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    (x+4)^2 = x^2 +8x +16 right? we can easily move back and forth between these equations since the are equal to each other...

  23. amistre64
    • 5 years ago
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    but what is usually missing in order to "complete" a sqaure is this: (x+___)^2 = x^2 +8x + ______

  24. amistre64
    • 5 years ago
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    we need 2 numbers that are exactly the same, that add to get 8. what numbers are they?

  25. anonymous
    • 5 years ago
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    4

  26. amistre64
    • 5 years ago
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    good :) 4+4 = 8, that will satisfy that middle term; now we multiply 4*4 to get the last term. 4^2 = 16 (x+4)^2 = x^2 +8x + 16 you see where we got it now?

  27. anonymous
    • 5 years ago
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    yeah

  28. amistre64
    • 5 years ago
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    x^2 + 6x + ____ what would we complete the square with here?

  29. anonymous
    • 5 years ago
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    (x+3)²=x²+6x+12?

  30. amistre64
    • 5 years ago
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    real close....look again. 3+3 = 6 3*3 != 12

  31. amistre64
    • 5 years ago
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    3*3 = ?

  32. anonymous
    • 5 years ago
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    9

  33. amistre64
    • 5 years ago
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    good :) we need to add 9 to "complete" the square what happens when you add a number to one side of an equation? what do we do to the other side?

  34. anonymous
    • 5 years ago
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    Add it on the other side

  35. amistre64
    • 5 years ago
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    Exactly :) Do you see where I did that in the problem? x^2 + y^2 +8 = 0 ; complete square for "y" x^2 + y^2 +8y +16 = 16

  36. amistre64
    • 5 years ago
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    i really should fix that typo lol

  37. anonymous
    • 5 years ago
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    Haha no. That's where I'm stuck.

  38. amistre64
    • 5 years ago
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    x^2 + [ y^2 +8y +____] = 0 ; complete square for "y" what number do we use to "complete" the square for y?

  39. anonymous
    • 5 years ago
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    16

  40. amistre64
    • 5 years ago
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    its always gonna be half the middle term and then square it

  41. amistre64
    • 5 years ago
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    8/2 = 4 -> 4^2 = 16 :) good

  42. amistre64
    • 5 years ago
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    x^2 + [y^2 +8y +16] = 0 + 16 ^^^ ^^^ we add 16 to both sides righ there right?

  43. anonymous
    • 5 years ago
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    Yeah

  44. amistre64
    • 5 years ago
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    then we just clean it up: x^2 + (y+4)^2 = 16

  45. amistre64
    • 5 years ago
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    which agrees with what we drew in the first place ;)

  46. anonymous
    • 5 years ago
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    Where did the 8y go then?

  47. amistre64
    • 5 years ago
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    remember we can move between the forms of a complete square? (y+4)^2 = y^2 +8y +16 we just use the one for the other.... they are identical in value, they only look different in form.

  48. anonymous
    • 5 years ago
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    Ohhh ok. I get it now!!

  49. amistre64
    • 5 years ago
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    once we got a "complete" square, we use it to clean up the equation ;)

  50. anonymous
    • 5 years ago
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    Ok, I finally understand this now! Thanks for helping me out!

  51. amistre64
    • 5 years ago
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    youre welcome :)

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