I'm trying to find the slope of an astroid. https://en.wikipedia.org/wiki/Astroid
Using $x = a\sin^3(t)$ and $y = a\cos^3(t)$, $dy(t)/dx(t) = -\tan(t)$. I expect $\tan(t) = y/x$, giving $dy/dx (x, y) = -y/x$, but this differs from the answer from implicit differentiation of $x^{2/3} + y^{2/3} = a^{2/3}$, which is $dy/dx(x, y) = -y^{1/3}/x^{1/3}$. Why is this? Is my expression for $\tan(t)$ wrong?