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Homework Help: Fairly simple trig question concerning calculus tools

  1. Jul 18, 2010 #1
    1. The problem statement, all variables and given/known data

    Well this is technically from a calculus problem but my question focuses only on the trig of the problem so I am posting it here. This is for graphing second degree equations with a nonzero xy

    2. Relevant equations
    Given:

    [tex] Ax^{2}+Bxy+Cy^{2}+Dx+Ey+F=0 [/tex]

    where [tex] B \neq 0 [/tex]

    Use the rotation of axes equations to find an equation where B=0. Equations to do so:

    [tex] x = X cos(\alpha) - Y sin(\alpha) [/tex]

    and

    [tex] y = X sin(\alpha) + Y cos(\alpha) [/tex]

    and alpha is given as:

    [tex] cot(2\alpha) = \frac{A-C}{B} [/tex]

    SO finally my question, how to solve for alpha, I think that I have just forgotten my trig or something here but an attempt I made looks like so:


    3. The attempt at a solution

    [tex] cot(2\alpha) = \frac {A-C}{B} [/tex]

    [tex] 2\alpha = cot^{-1} (\frac {A-C}{B}) [/tex]

    [tex] \alpha = \frac {cot^{-1}(\frac{A-C}{B})}{2} [/tex]

    and if that is correct that is all good and all but I don't remember how to solve for an inverse cotangent or how to enter it into a graphing calc so if I am right with my equation above then can someone re-enlighten me on this?

    Thanks!
     
  2. jcsd
  3. Jul 18, 2010 #2

    vela

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    Recall that cotangent and tangent are reciprocals.
     
  4. Jul 18, 2010 #3
    ha right ok so just so I know that I am right here

    [tex] \frac {cot^{-1} (\frac{A-C}{B})}{2} = \frac {tan (\frac{A-C}{B})}{2} [/tex]

    right?
     
  5. Jul 18, 2010 #4

    vela

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    No, you want to use the fact that

    [tex]\cot 2\alpha = \frac{1}{\tan 2\alpha} = \frac{A-C}{B}[/tex]
     
  6. Jul 18, 2010 #5
    ok so then if:

    [tex] \frac {1}{tan(2\alpha)} = \frac {A-C}{B} [/tex]

    then

    [tex] tan(2\alpha) = \frac {B}{A-C} [/tex]

    so

    [tex] 2\alpha = tan^{-1} (\frac{B}{A-C}) [/tex]

    [tex] \alpha = \frac {tan^{-1}(\frac{B}{A-C})}{2} [/tex]

    yeah?
     
  7. Jul 18, 2010 #6

    vela

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    Yup.
     
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