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Math Help - Finding angle of rotation. Test this Monday. I missed school. Seriously need help.

  1. #1
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    Finding angle of rotation. Test this Monday. I missed school. Seriously need help.

    Ok. Here is what i do know:
    in an equation: Ax + Bxy + Cy + Dx + Ey + F = 0
    i need to eliminate the Bxy part, and find an angle of rotation.
    cot(2Ɵ) =(a-c)/b
    from there i can find Ɵ and create a new x',y' equation without a Bxy part.

    Here is my question:
    when there are 2 possible values for Ɵ, which do i use?
    this has come up on my homework a lot.

    Here's an example:
    xy+1 = 0
    cot(2Ɵ) = 0
    2Ɵ could either be (pi/2) or (3pi/2)
    Ɵ could either be (pi/4) or (3pi/4)

    another example:
    2x - 3xy - 2y + 10 = 0
    cot(2Ɵ) = (-4/3)
    by constructing a right triangle i can figure that cos(2Ɵ) can either be (-4/5) or (4/5)

    from what i can tell, sometimes either value of Ɵ will result in me making the same graph, but the x'y' equation still differs. Are both answers always equally correct? If so, then which way is the general way expected on a math test, because some answers wont ask for a graph, just the x'y' equation.

    what do i do and why?
    My class is a very high level honors class, and i am not going to be able to get a chance to ask the teacher before the test on monday. Thanks so much in advance.
    Last edited by brandon; February 20th 2010 at 03:49 PM.
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  2. #2
    MHF Contributor

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    Quote Originally Posted by brandon View Post
    Ok. Here is what i do know:
    in an equation: Ax + Bxy + Cy + Dx + Ey + F = 0
    i need to eliminate the Bxy part, and find an angle of rotation.
    cot(2Ɵ) =(a-c)/b
    from there i can find Ɵ and create a new x',y' equation without a Bxy part.

    Here is my question:
    when there are 2 possible values for Ɵ, which do i use?
    this has come up on my homework a lot.

    Here's an example:
    xy+1 = 0
    cot(2Ɵ) = 0
    2Ɵ could either be (pi/2) or (3pi/2)
    Ɵ could either be (pi/4) or (3pi/4)

    another example:
    2x - 3xy - 2y + 10 = 0
    cot(2Ɵ) = (-4/3)
    by constructing a right triangle i can figure that cos(2Ɵ) can either be (-4/5) or (4/5)
    Didn't you just say -4/3? Did you mean -4/3 and 4/3?

    from what i can tell, sometimes either value of Ɵ will result in me making the same graph, but the x'y' equation still differs. Are both answers always equally correct? If so, then which way is the general way expected on a math test, because some answers wont ask for a graph, just the x'y' equation.
    The difference is whether you are rotating clockwise or counterclockwise and that will result in swapping x' and y' axes.

    [/quote]what do i do and why?
    My class is a very high level honors class, and i am not going to be able to get a chance to ask the teacher before the test on monday. Thanks so much in advance.[/QUOTE]
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  3. #3
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    Joined
    Feb 2010
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    thanks for the reply, but unfortunately it didnt really help.

    Didn't you just say -4/3? Did you mean -4/3 and 4/3?
    no, i meant what i said:
    cot(2Ɵ) = (-4/3)
    cos(2Ɵ) can either be (-4/5) or (4/5)
    check the attached image for clarification

    The difference is whether you are rotating clockwise or counterclockwise and that will result in swapping x' and y' axes.
    if i choose cos(2Ɵ) = (-4/5) that puts 2Ɵ into the second quadrant and Ɵ into the first

    if i choose cos(2Ɵ) = (4/5) that puts 2Ɵ into the fourth quadrant and Ɵ into the second

    neither involves Ɵ being negative therefore both are standard counterclockwise rotations

    please correct me if i am wrong

    since both involve standard counterclockwise rotations, which way is correct?


    also, in case i ever do have to graph a rotation involving a negative angle, what did you mean by swapping the x' and y' axes

    thanks
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