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Math Help - existence proof, how to start?

  1. #1
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    existence proof, how to start?

    Given the equation C\sin (x + \alpha) = A\sin x + B\cos x, I need to prove that C and \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
    Super Member Failure's Avatar
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    Quote Originally Posted by Dementiy View Post
    Given the equation C\sin (x + \alpha) = A\sin x + B\cos x, I need to prove that C and \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 C :=\sqrt{A^2+B^2}, and divide through to get

    \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 \color{red}\cos(\alpha)=\frac{A}{C}, and that \color{blue}\sin(\alpha)=\frac{B}{C}, which is always possible to satisfy because \left(\frac{A}{C}\right)^2+\left(\frac{B}{C}\right  )^2=1, given our choice of C.
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