To understand how the two standard ways to write the general solution to a harmonic oscillator are related. There are two common forms for the general solution for the position of a harmonic oscillator as a function of time t:

x(t)=Acos(ωt+ϕ) and
x(t)=Ccos(ωt)+Ssin(ωt).

Either of these equations is a general solution of a second-order differential equation (F⃗ =ma⃗ ); hence both must have at least two--arbitrary constants--parameters that can be adjusted to fit the solution to the particular motion at hand. (Some texts refer to these arbitrary constants as boundary values.)

Required:
a. Find analytic expressions for the arbitrary constants C.
b. Find analytic expressions for the arbitrary constants A and ϕ.