Sinusoids are fundamental in a variety of ways.
One reason for the importance of sinusoids is that they are fundamental in physics. Anything that resonates or oscillates produces quasi-sinusoidal motion. See simple harmonic motion in any freshman physics text for an introduction to this topic.
Another reason sinusoids are important is that they are eigenfunctionsof linear systems (which we'll say more about later). This means that they are important for the analysis of filters such as reverberators, equalizers, certain (but not all) ''effects'', etc.
Perhaps most importantly, from the point of view of computer music research, is that the human ear is a kind of spectrum analyzer. That is, the chochlea of the inner ear physically splits sound into its (near) sinusoidal components. This is accomplished by the basilar membrane in the inner ear: a sound wave injected at the oval window(which is connected via the bones of the middle ear to the ear drum), travels along the basilar membrane inside the coiled cochlea. The membrane starts out thick and stiff, and gradually becomes thinner and more compliant toward its apex (the helicotrema). A stiff membrane has a high resonance frequency while a thin, compliant membrane has a low resonance frequency (assuming comparable mass density, or at least less of a difference in mass than in compliance). Thus, as the sound wave travels, each frequency in the sound resonates at a particular place along the basilar membrane. The highest frequencies resonate right at the entrance, while the lowest frequencies travel the farthest and resonate near the helicotrema. The membrane resonance effectively ''shorts out'' the signal energy at that frequency, and it travels no further. Along the basilar membrane there are hair cells which ''feel'' the resonant vibration and transmit an increased firing rate along the auditory nerve to the brain. Thus, the ear is very literally a Fourier analyzer for sound, albeit nonlinear and using ''analysis'' parameters that are difficult to match exactly. Nevertheless, by looking at spectra (which display the amount of each sinusoidal frequency present in a sound), we are looking at a representation much more like what the brain receives when we hear.