I have learned how to compute a tangent cone if we have a given polynomial in the following way: consider the part of the polynomial that is of minimal homogeneous degree. For example if $f=x^2+xy-y^3$, then consider $f_m=x^2+xy$=$x(x+y)$ and then the tangent cone is given by setting $f_m=0$, so we have the lines $x=0$ and $y=-x$.
Now, my question is, given an ideal of a variety, $I(V)$, how does one recognize the ideal of a tangent cone? For example, if $I(V)=(xy,yz,zx)$ a subset of $k[x,y,z]$ (which the variety is the 3 coordinate axis in $R^3$), what is the tangent cone of the variety $V$? Then, how would one determine what the ideal of that cone is?