Let $C: y^2=x^3$ be a singular elliptic curve over a field K and define $C_{ns}(K)=C(K)/{0,0}$ the set of non singular points of C including $P_\infty$. I've proved that $C_{ns}$ is a group with the usual addition of points in an elliptic curve. I am asked to define a group isomorphism $\phi: C_{ns}(K) \to (K,+)$.
I noticed that points in $C_{ns}(K)$ correspond to lines through the origin so I tried to define $\phi(x,y)=\frac{y}{x}$, sending a point to the slope of the line that passes through the origin and the point, but I am not sure if this is an homomorphism.
The affine points of $C$ are the $(t^2,t^3)$ (where $t$ corresponds to your $\phi(x,y)=y/x$). Three distinct points on $C$ are collinear iff $$\left|\matrix{t_1^2&t_1^3&1\\t_2^2&t_2^3&1\\t_3^2&t_3^3&1\\}\right|=0,$$ which is equivalent to $$\frac1{t_1}+\frac1{t_2}+\frac1{t_3}=0.$$ If we map $(t^2,t^3)$ to $1/t$, we therefore get a group isomorphism (if in addition we map the point at infinity to $0$) from $C_{ns}(K)$ to $K$.