Harmonic motion, finding analytic expressions for constants

[Linus Pauling]

1. Find analytic expressions for the arbitrary constants A and phi in Equation 1 (found in Part A) in terms of the constants C and S in Equation 2 (found in Part B), which are now considered as given parameters.
Express the amplitude A and phase phi (separated by a comma) in terms of C and S.




2. x(t) = Acos(omega*t + phi_
x(t) = Ccos(omega*t) + Ssin(omega*t)




3. I have found the following:

C = Acos(phi)
S = -Asin(phi)
C^2 + S^2 = A^2

And I am lost at this point.

 

[Linus Pauling]

Nevermind, I figured it out.

 

[kerimek]

How do I continue with finding analytic expressions for A and phi?

I would suggest using trigonometric identities to solve for A and phi. From the equations given, we can see that A represents the amplitude of the harmonic motion and phi represents the initial phase of the motion. We can use the trigonometric identity cos^2(x) + sin^2(x) = 1 to simplify the equation C^2 + S^2 = A^2. This gives us:

A^2 = C^2 + S^2 = (Acos(phi))^2 + (-Asin(phi))^2 = A^2(cos^2(phi) + sin^2(phi))

We can then divide both sides by A^2 and take the square root to solve for A:

A = √(C^2 + S^2)

Next, we can use the trigonometric identity tan(x) = sin(x)/cos(x) to solve for phi. We can rearrange the given equations to get:

tan(phi) = -S/C

This gives us:

phi = arctan(-S/C)

Therefore, the analytic expressions for A and phi in terms of C and S are:

A = √(C^2 + S^2)
phi = arctan(-S/C)

These expressions allow us to find the values of A and phi for any given values of C and S, and thus fully describe the harmonic motion given by the equations.