Solution to a slope field through given points

Question

I want to find solutions to a slope field given certain initial conditions. Say I have a slope field of the form: $$\frac{dx}{dt} = \cos(z)$$ $$\frac{dy}{dt} = \sin(z)$$ $$\frac{dz}{dt} = f(x,y,z)$$ Given $x_0, y_0, x_1, y_1$, I want to find $z_0, z_1$ such that there is a solution to the slope field passing through $(x_0, y_0, z_0)$ and $(x_1, y_1, z_1)$. What methods exist to solve problems of this sort? My end goal is to solve problems of this sort numerically with both consistency and speed on a computer, but theoretical insight is useful too.