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Thread: existence proof, how to start?

  1. #1
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    existence proof, how to start?

    Given the equation $\displaystyle C\sin (x + \alpha) = A\sin x + B\cos x$, I need to prove that $\displaystyle C$ and $\displaystyle \alpha$ exist given any real A and B, C >= 0. On a previous problem I proved the converse, that A and B exist, but I'm not sure where to start on this one.
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  2. #2
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    Quote Originally Posted by Dementiy View Post
    Given the equation $\displaystyle C\sin (x + \alpha) = A\sin x + B\cos x$, I need to prove that $\displaystyle C$ and $\displaystyle \alpha$ exist given any real A and B, C >= 0. On a previous problem I proved the converse, that A and B exist, but I'm not sure where to start on this one.
    You can define $\displaystyle C :=\sqrt{A^2+B^2}$, and divide through to get

    $\displaystyle \sin(x+\alpha)=\sin(x)\cdot{\color{red}\cos(\alpha )}+\cos(x)\cdot{\color{blue}\sin(\alpha)}={\color{ red}\frac{A}{C}}\cdot\sin(x)+{\color{blue}\frac{B} {C}}\cdot\cos(x)$

    Comparing what you've got you must require that $\displaystyle \color{red}\cos(\alpha)=\frac{A}{C}$, and that $\displaystyle \color{blue}\sin(\alpha)=\frac{B}{C}$, which is always possible to satisfy because $\displaystyle \left(\frac{A}{C}\right)^2+\left(\frac{B}{C}\right )^2=1$, given our choice of $\displaystyle C$.
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