# Parametrization of the implicit curve $F(x,y)=a_1\cos x+a_2\sin x+a_3\cos y+a_4\sin y-b=0$

I am trying to find a parametrization for the curve defined implicitly by $$F(x,y)=a_1\cos x+a_2\sin x+a_3\cos y+a_4\sin y-b=0,$$ where $a_1$, $a_2$, $a_3$, $a_4$ and $b$ are constants satisfying $a_1^2+a_2^2+a_3^2+a_4^2+b^2=1$, but I do not know what to try. I would like to be able to do it by myself, so I am looking mostly for hints on how to perform this: what methods I need to apply and what steps should I follow.

As a first step, you can combine $$\begin{gathered} {a_1}\cos x + {a_2}\sin x = \sqrt {{a_1}^2 + {a_2}^2} \left( {\frac{{{a_1}}}{{\sqrt {{a_1}^2 + {a_2}^2} }}\cos x + \frac{{{a_2}}}{{\sqrt {{a_1}^2 + {a_2}^2} }}\sin x} \right) \\ = \sqrt {{a_1}^2 + {a_2}^2} \sin (\theta + x) \\ \end{gathered}$$ for some constant angle $\theta$ (where $\theta = {\text{atan}}2({a_1},{a_2})$). You can do similarly for ${a_3}\cos y + {a_4}\sin y$. This will give you just one variable $x$ and one variable $y$ and an easier equation.