I'm reading Hibbeler 14th edition for dynamics, in chapter 13 part 7(13.7) it's talking about gravitational force, and at this point it is talking about the equation of free-flight trajectory.
Base on what I know, the answer to this is: $\xi = C_1\sin(\theta) + C_2\cos(\theta) + \dfrac{GM_e}{h^2}$
But this is the answer the book is giving me: $\xi = C * \cos(\theta - \phi) + \dfrac{GM_e}{h^2}$
I don't understand why this is true.
$$C\cos(\theta-\phi) = C\cos(\phi)\cos(\theta) + C\sin(\phi)\sin(\theta) = c_1\cos(\theta) + c_2\sin(\theta).$$
It's just a different way of parametizing the arbitrary constants. You can use $C$ and $\phi$ or $c_1$ and $c_2$.
You can relate them by $c_2/c_1 = \tan\phi$ and $c_1^2 + c_2^2 = C^2$. One system expresses the coefficients of $cos\theta$ and $\sin\theta$ in Cartesian coordinates, the other in polar.