Circle Lattice

We can start by finding the first positive root in $x$ by letting $y = 0$ .
$\sin(x) \sin(x) = 0.9.$ so that $x = \sin^{(-1)}(\sqrt{0.9})$ . To find the diameter, $D$ , we seek the next positive root in $x$ .
$\sin(x+D) \sin(x+D) = 0.9,$ with $x$ given previously. Using an identity,
$\sin(x) \cos(D) + \cos(x) \sin(D) = \sqrt{0.9}.$ Substituting $x$ , we get
$\sqrt{0.9} \cos(D) + \sqrt{0.1} \sin(D) = \sqrt{0.9}.$
Solving this equation yields the answer $x \approx 0.6435$ .