How were amplitudes of the $\cos$ and $\sin$ chosen?

Let’s concentrate on the important part, which is of the form $$ f(x)=a\cos x+b\sin x $$ which we want to express as $$ f(x)=A(\cos\varphi\cos x+\sin\varphi\sin x) $$ A necessary (and sufficient) condition is that $$ A\cos\varphi=a,\qquad A\sin\varphi=b $$ and therefore $a^2=A^2\cos^2\varphi$, $b^2=A^2\sin^2\varphi$. Hence $$ A^2=a^2+b^2 $$ We want $A>0$ (not necessary, but convenient), so we get $$ A=\sqrt{a^2+b^2},\quad \cos\varphi=\frac{a}{A},\quad \sin\varphi=\frac{b}{A} $$ The last two requirements can be fulfilled, because $(a/A,b/A)$ is a point on the unit circle.