I understand how $A\sin(x+\alpha)$ is expanded to produce the form $a\sin(x)+b\cos(x)$, which means that $\tan(\alpha)=\frac{b}{a}$ and $A^2=a^2+b^2$, but I just can't wrap my head aroud why a suitable $\alpha$ and $A$ can be found for ANY coefficients of $\sin(x)$ and $\cos(x)$. For instance, if you choose a certain value of $\alpha$, for which the coefficient of $\sin(x)$ works with a given $A$, then the coefficient of $\cos(x)$ is already determined. Now what if you changed that coefficient a little bit?
I'm just finding it hard to grasp how an $A$ and $\alpha$ can be found for any $a$ and $b$…
Best Answer
$$a\sin x+b \cos x=\sqrt{a^2+b^2}\left(\frac{a}{\sqrt{a^2+b^2}}\sin x+\frac{b}{\sqrt{a^2+b^2}}\cos x\right)=A\sin(x+\alpha)$$ where $A=\sqrt{a^2+b^2}$ and $\alpha= \arcsin\frac{b}{\sqrt{a^2+b^2}}$