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Math Help - How to eliminate angle from equations.

  1. #1
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    How to eliminate angle from equations.

    Hi there, I have to eliminate angle φ from equations:
    How to eliminate angle from equations.-1.png
    And transform to the equation of circle:
    How to eliminate angle from equations.-2.png
    Please help.
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  2. #2
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    Re: How to eliminate angle from equations.

    Hello, blueye89!

    There is a typo . . . one of the equations must have a minus-sign.


    I have to eliminate \theta from these equations:

    . . x \:=\:\tfrac{15}{2} + \tfrac{3}{2}\cos2\theta + 2\sin2\theta

    . . y \:=\: \tfrac{3}{2}\sin2\theta\,{\color{red}-}\,2\cos2\theta
    n . . . . . . . . . . . {\color{blue}\uparrow}

    And transform to the equation of circle: . \left(x-\tfrac{15}{2}\right)^2 + y^2 \:=\:\left(\tfrac{5}{2}\right)^2

    We have: . \begin{Bmatrix}x-\frac{15}{2} &=& 2\sin2\theta + \frac{3}{2}\cos2\theta \\ \\[-4mm] y &=& \frac{3}{2}\sin2\theta - 2\cos2\theta \end{Bmatrix}


    Square both equations: . \begin{Bmatrix}(x-\frac{15}{2})^2 &=& 4\sin^2\!2\theta + 6\sin2\tyheta\cos2\theta + \frac{9}{4}\cos^2\!2\theta \\ \\[-3mm] y^2 &=& \frac{9}{4}\sin^2\!2\theta - 6\sin2\theta\cos\theta + 4\cos^2\!2\theta \end{Bmatrix}

    Add: . \left(x-\tfrac{15}{2}\right)^2 + y^2 \;=\;\tfrac{25}{4}\cos^2\!2\theta + \tfrac{25}{4}\sin^2\!2\theta

    n . . . \left(x-\tfrac{15}{2}\right)^2 + y^2 \;=\;\tfrac{25}{4}\underbrace{(\sin^2\!2\theta + \cos^2\!2\theta)}_{\text{This is 1}} \;=\;\tfrac{25}{4}
    n . . . \left(x-\tfrac{15}{2}\right)^2 + y^2 \;=\;\left(\tfrac{5}{2}\right)^2
    Thanks from blueye89
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  3. #3
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    Re: How to eliminate angle from equations.

    Yes my friend. Sorry,my mistake, there is an minus in second equation. Thank you very much.
    Last edited by blueye89; December 18th 2012 at 05:49 PM.
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