So I am studying spring mass systems, and I'm confused by why the solution, which is complex, becomes real. Such that
$$x(t) = C_1\cos(\omega t)+iC_2\sin(\omega t) = C_1\cos(\omega t)+C_2\sin(\omega t)$$
Why does the imaginary term just become an arbitrary constant?
Also the general solution is also simplified to:
How do you get that result?
I tried drawing sine and cosine waves to correlate them, but I don't see it. I also looked at some trig identities, but no avail.