A nice trig identity #
\[ \cos(x+y)=\cos(x)\cos(y)-\sin(x)\sin(y) \]
$$\cos(\omega x+\varphi)=\cos(\omega x)\cos(\varphi)-\sin(\omega x)\sin(\varphi)$$ $$R\cos(\omega x+\varphi)=R\cos(\omega x)\cos(\varphi)-R\sin(\omega x)\sin(\varphi)$$ $$R\cos(\omega x+\varphi)=\left(R\cos(\varphi)\right)\cos(\omega x)+\left(-R\sin(\varphi)\right)\sin(\omega x)$$ $$a\triangleq R\cos(\varphi)$$ $$b\triangleq-R\sin(\varphi)\implies-b=R\sin(\varphi)$$ $$R\cos(\omega x+\varphi)=a\cos(\omega x)+b\sin(\omega x)$$ $$\sqrt{a^2+b^2}=\sqrt{\left(R\cos(\varphi)\right)^2+\left(-R\sin(\varphi)\right)^2}=\sqrt{R^2\left(\cos^2(\varphi)+\sin^2(\varphi)\right)}=\sqrt{R^2}=|R|$$ $$z\triangleq a+-bi=R\cos(\varphi)+R\sin(\varphi)i$$ $$\mathfrak{R}\{z\}=a=R\cos(\varphi)\quad\text{and}\quad\mathfrak{I}\{z\}=-b=R\sin(\varphi)$$ $$|R|=\sqrt{a^2+b^2}=|z|$$ $$\varphi=\arg(z)$$ $$-$$ $$R\cos(\omega x+\varphi)=a\cos(\omega x)+b\sin(\omega x)$$ $$-$$ $$\text{Non-unique solution to convert sum of same-frequency sinusoids to cosine in the above format:}$$ $$\quad R=\sqrt{a^2+b^2}\quad\text{and}\quad\varphi=\operatorname{{atan}2}(-b,a)$$ $$-$$ $$\text{Unique solution to convert cosine to sum of same-frequency sinusoids in the above format:}$$ $$\quad a=R\cos(\varphi)\quad\text{and}\quad b=-R\sin(\varphi)$$