I have a system of two equations, involving trigonometric terms of two variables $\theta$ and $\phi$:

$$ d\_1 = a\_1\sin\theta \cos\phi + b\_1\sin\phi + c\_1\cos\theta\cos\phi $$
$$ d\_2 = a\_2\sin\theta \cos\phi + b\_2\sin\phi + c\_2\cos\theta\cos\phi $$

$d\_1$, $d\_2$, $a\_1$, $a\_2$, $b\_1$, $b\_2$, $c\_1$ and $c\_2$ being known, real constants.

(if we assume $\theta$ and $\phi$ to be spherical angles, you will probably recognize spherical harmonics of degree 1 at work here).

Other than somewhat brute-force numerical resolution methods, is there a way to more or less elegantly solve this system of equations in $\theta$ and $\phi$ ?

Thanks!