# Thread: existence proof, how to start?

1. ## 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.

2. Originally Posted by Dementiy
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 $C$.