Very Challenging Oscillations Problem!!?

Question

There are two common forms for the general solution for the position of a harmonic oscillator as a function of time t:

x(t) = A cos(omega*t + phi) and

x(t) = C cos(omega*t) + S sin(omega*t)

Either of these equations is a general solution of a second-order differential equation (F = ma); hence both must have at least two--arbitrary constants--parameters that can be adjusted to fit the solution to the particular motion at hand.

1) Find analytic expressions for the arbitrary constants C and S in Equation 2 (found in Part B) in terms of the constants A and phi in Equation 1 (found in Part A), which are now considered as given parameters. Give your answers for the coefficients of cos(omega*t) and sin(omega*t), separated by a comma. Express your answers in terms of A and phi.

Answer 1

Apparently, the only "Physics" in this question is to recognize that the two equations share the same initial conditions - at t = 0, the position x and the velocity x'.

Substitute t = 0 in both equations, and in the derivatives of both equations. You will get equations without the t in a sine or cosine. The rest is simple algebra without trigonometry. Remember that cos phi and sin phi are OK in the equations since they are just constants; it is the cos (....t...) and sin (....t...) that are problems - and you won't have them if you substitute t = 0.