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Math Help - Help With Trig Equation

  1. #1
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    Help With Trig Equation

    if: xsin(A) + ycos(A) = R and xcos(A) - ysin(A)= Q

    Find the value of (x^2) + (y^2)

    thanks
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  2. #2
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    Hello, white_cap!

    If . \begin{array}{ccc}x\sin(A) + y\cos(A) &= &R \\ x\cos(A) - y\sin(A)& =& Q\end{array}

    find the value of: . x^2 + y^2

    Square both equations:

    [x\sin(A)\;+\;y\cos(A)]^2 \;=\;R^2 \quad \Rightarrow . . x^2\sin^2(A)\;+\;2xy\sin(A)\cos(A)\;+\;y^2\cos^2(A  ) \;=\;R^2

    [x\cos(A)\;-\;y\sin(A)]^2\;=\;Q^2 \quad\Rightarrow . . x^2\cos^2(A)\;-\;2xy\sin(A)\cos(A)\;+\;y^2\sin^2(A) \;=\;Q^2


    Add the equations: . x^2\!\cdot\!\underbrace{\left[\sin^2(A) + \cos^2(A)\right]}_{\text{This is 1}} +\: y^2\!\cdot\!\underbrace{\left[\sin^2(A) + \cos^2(A)\right])}_{\text{This is 1}} \;=\;R^2+Q^2


    Therefore: . x^2 + y^2 \;=\;R^2+Q^2

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  3. #3
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    Wow that was brilliant. I understand it, but how did (would) you ever think of adding both equations, unless you wrote them above each other and realised that it would 'work out' nicely?
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  4. #4
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    Hello again, white_cap!


    It's one of those thing we learn to watch for.

    When I see both \sin\theta and \cos\theta, that technique come to mind


    It turns up regularly in Parametric Equations.

    Example: . \begin{array}{ccc}x & = & 3\sin\theta \\ y & = & 2\cos\theta\end{array}

    Eliminate the parameter \theta.
    Get an equation involving x and y (only).


    We have: . \begin{Bmatrix}\dfrac{x}{3} \:=\:\sin\theta \\ \\ \dfrac{y}{2} \:=\:\cos\theta \end{Bmatrix}
    Square the equations: . \begin{Bmatrix}\dfrac{x^2}{9} \:=\:\sin^2\theta \\ \\ \dfrac{y^2}{4} \:=\:\cos^2\theta \end{Bmatrix}

    Add the equations: . \frac{x^2}{9} + \frac{y^2}{4} \:=\:\underbrace{\sin^2\theta + \cos^2\theta}_{\text{This is 1}}

    Therefore: . \frac{x^2}{9} + \frac{y^2}{4} \:=\:1 . . . . There!

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