Sinusoids are really very desirable functions in many problems of Signal Processing, including yours. Due to the high importance of these properties, I'll proclaim them in brief:
1) Sinusoids (with different frequencies) are orthogonal (it may be proved both in time or frequency domain or with Parseval equation). It allows to evaluate decomposition coefficients much easier, than in the case of orthogonality absence (Remember the generalized Fourier series).
2) When analyzing the arbitrary linear chain with harmonic input signals we can use symbolic method (method of complex amplitudes)
3) Sinusoidal signal never changes its analytical form when passing through any linear system. It means, that the output reaction is sinusoidal as well, only amplitude and initial phase changes. They may be accurately computed by method of complex amplitudes.
4) Sinusoids are smooth, infinitely differentiated and easily tranformed in other trigonometric functions (remember formulas of double argument). Therefore they are suitable for approximating smooth, slowly-changing processes.
5) They have the best frequency localization (Fourier spectrum of sinus is just 1 harmonic on carrier fequency).
The main disatvantage is that they don't have time localization. They are global. Therefore in analyzing non-stationary processes they're usually neglected and wavelets, EMD and SSA are preferred.
With respect,
Dmitrij[/b]