$\overline{){\mathbf{c}}{\mathbf{o}}{\mathbf{s}}{\mathbf{}}{\mathbf{(}}{\mathbf{a}}{\mathbf{}}{\mathbf{+}}{\mathbf{}}{\mathbf{b}}{\mathbf{)}}{\mathbf{}}{\mathbf{=}}{\mathbf{}}{\mathbf{c}}{\mathbf{o}}{\mathbf{s}}{\mathbf{}}{\mathbf{a}}{\mathbf{\xb7}}{\mathbf{c}}{\mathbf{o}}{\mathbf{s}}{\mathbf{}}{\mathbf{b}}{\mathbf{}}{\mathbf{-}}{\mathbf{}}{\mathbf{s}}{\mathbf{i}}{\mathbf{n}}{\mathbf{}}{\mathbf{a}}{\mathbf{\xb7}}{\mathbf{s}}{\mathbf{i}}{\mathbf{n}}{\mathbf{}}{\mathbf{b}}}$

$\overline{){\mathbf{c}}{\mathbf{o}}{{\mathbf{s}}}^{{\mathbf{2}}}{\mathbf{}}{\mathbf{\theta}}{\mathbf{+}}{\mathbf{}}{\mathbf{s}}{\mathbf{i}}{{\mathbf{n}}}^{{\mathbf{2}}}{\mathbf{}}{\mathbf{\theta}}{\mathbf{=}}{\mathbf{1}}}$

$\overline{){\mathbf{t}}{\mathbf{a}}{\mathbf{n}}{\mathbf{}}{\mathbf{\theta}}{\mathbf{=}}\frac{\mathbf{s}\mathbf{i}\mathbf{n}\mathbf{}\mathbf{\theta}}{\mathbf{c}\mathbf{o}\mathbf{s}\mathbf{}\mathbf{\theta}}}$

*A) *

*x*(*t*) = *A*cos(*ω**t *+ *ϕ*)

$\begin{array}{rcl}\mathbf{x}\mathbf{\left(}\mathbf{t}\mathbf{\right)}& \mathbf{=}& \mathbf{A}\mathbf{}\mathbf{c}\mathbf{o}\mathbf{s}\mathbf{}\mathbf{(}\mathbf{\omega}\mathbf{t}\mathbf{}\mathbf{+}\mathbf{}\mathbf{\varphi}\mathbf{)}\\ & \mathbf{=}& \mathbf{A}\mathbf{(}\mathbf{c}\mathbf{o}\mathbf{s}\mathbf{}\mathbf{\omega}\mathbf{t}\mathbf{\xb7}\mathbf{c}\mathbf{o}\mathbf{s}\mathbf{}\mathbf{\varphi}\mathbf{-}\mathbf{s}\mathbf{i}\mathbf{n}\mathbf{}\mathbf{\omega}\mathbf{t}\mathbf{\xb7}\mathbf{s}\mathbf{i}\mathbf{n}\mathbf{}\mathbf{\varphi}\\ & \mathbf{=}& \mathbf{(}\mathbf{A}\mathbf{c}\mathbf{o}\mathbf{s}\mathbf{}\mathbf{\varphi}\mathbf{)}\mathbf{c}\mathbf{o}\mathbf{s}\mathbf{}\mathbf{\omega}\mathbf{t}\mathbf{+}\mathbf{(}\mathbf{-}\mathbf{A}\mathbf{s}\mathbf{i}\mathbf{n}\mathbf{}\mathbf{\varphi}\mathbf{)}\mathbf{s}\mathbf{i}\mathbf{n}\mathbf{}\mathbf{\omega}\mathbf{t}\end{array}$

Objective: 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*)=*A *cos(*ω**t*+*ϕ*) and

*x*(*t*)=*C *cos(*ω**t*)+*S *sin(*ω**t*).

Either of these equations is a general solution of a second-order differential equation (*F*⃗ =*m**a*⃗ ); 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*.)

A)

Find analytic expressions for the arbitrary constants *C* and *S* in Equation 2 (found in Part B) in terms of the constants *A* and *ϕ* in Equation 1 (found in Part A), which are now considered as given parameters.

Give your answers for the coefficients of cos(*ω**t*) and sin(*ω**t*), separated by a comma. Express your answers in terms of *A* and *ϕ*.

B)

Find analytic expressions for the arbitrary constants *A* and *ϕ* 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 *ϕ* (separated by a comma) in terms of *C* and *S*.

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