What is the value of y in $x\left(\tan\left(\sqrt{x^{2}+y^{2}}\right)\right)-y=0$

Question

What is the value of y in, $$x\left(\tan\left(\sqrt{x^{2}+y^{2}}\right)\right)-y=0$$, and in general how would you solve, sin(x+y)=sqrt(y), for y?

Answer 1

If you use polar coordinates $x=r \cos(t)$, $y=r \sin(t)$, you end with the equation $$r \tan (r) \cos (t)=r \sin (t)$$ the only non-trivial solution being $r=t$ which gives the equation of a spiral.

Otherwise, let $x=k y$ and, for a given value of $k$, you would have $$y=\frac{\cot ^{-1}(k)}{\sqrt{k^2+1}}$$