"An oscillation in a system is given by $\displaystyle x = \frac{4}{100} e^{- \frac{1}{100} t} \sin{(12t)} $. Write this in the form

$\displaystyle x = {\rm Re} \left( c e^{\alpha + i \beta} \right) $."

So I have:

$\displaystyle \begin{aligned}

x & = \frac{4}{100}e^{\frac{-1}{100}t}\sin(12t)\\

& = {\rm Re}\left(ce^{\alpha+i\beta}\right)\\

& = {\rm Re}\left(ce^{\alpha}e^{i\beta}\right)\\

& = {\rm Re}\left(ce^{\alpha}\left[\cos\beta+i\sin\beta\right]\right)\\

& = ce^{\alpha}\cos\beta\\

\frac{4}{100}e^{\frac{-1}{100}t}\sin(12t) & = ce^{\alpha}\sin\left(\beta-\frac{\pi}{2}\right)

\end{aligned} $

By comparing respective "components", I propose that

$\displaystyle c = \frac{4}{100}, \; \alpha = \frac{-1}{100} t , \; \text{ and } 12t = \beta - \frac{\pi}{2} , \; \text{ So, } \beta = 12t + \frac{\pi}{2} $

However, I'm not sure of the logically justification behind this. Why is OK to compare the respective components like this? For instance, is it impossible that one could transform both the exponential and the sine parts, so that they have different parameters, but their product remains unchanged, thereby introducing multiple solutions to the problem?