Significance of Sinusoidal signals
Significance of (pure) sinusoidals in engineering (summary):

1- Periodic signals: Pure sine-waves, represent a useful and simple tool in modeling various physical phenomena, such as standing waves, rotations, alternating currents etc. 

2- Fourier analysis: Periodic signals of any engineering sort (not necessarily of pure sine waves) are conveniently decomposed into pure sine waves acording to Fourier series analysis.

3- Linear Systems: strictly speaking complex exponentials $e^{j\omega t}$, are eigenfunctions of LTI systems but sine waves are deeply related. More specifically, the frequency of the sinusoid is preseverd but the amplitude and phase are changed by LTI systems, according to the system's frequency response $H(\omega)$. For instance, for a zero phase system (a symmetric noncausal impulse response) where $H(\omega)$ is real, then the pure sinusoids will also display the eigenfunction property.

4- Nonperiodic Signals: By the extension of Fourier series analysis to integral analysis of continuum frequencies, almost any practical signal is analysed by periodic sine waves.