homework and exercises - Harmonic oscillator differential equation solution

My math book explains how to solve second order equations like :

$$\ddot{x} + \omega^2x = 0$$

but I end up with the general solution :

$$A\cos(\omega t) + iB\sin(\omega t).$$

Now my physics book says the solution is

$$\rho\cos(\omega t + \phi)$$

How can I get there from the general solution?

Answer

Using the sum rule for cosines, we find

$$\rho \cos(\omega t + \phi) = \rho \cos(\phi) \cos(\omega t) - \rho \sin(\phi) sin(\omega t).$$ So we see that $\rho \cos(\omega t + \phi)$ is the same as $A\cos(\omega t) + iB\sin(\omega t)$ when $$A = \rho \cos(\phi)$$ and $$B = i\rho \sin(\phi).$$