I belive you and the word harsh wouldn't even have crossed my mind if you hadn't mentioned it.
Sine and cosine waves whose frequencies are multiples of a fundamental frequency, form an orthonormal basis w.r.t. the inner product given by integrating f(x)ĝ(x)dx over one period of the sine (or cosine) wave whose frequency is the fundamental frequency. In practical signal processing, this means that we can separately consider the contribution of each frequency to the total energy of a signal. Technically, the above only explains Fourier series, but the Fourier transform is just the limit case when the fundamental frequency is made arbitrarily small (re-scaled by a factor of the size of Dirac's delta at 0, of course).
One major benefit of the sin/cos basis is that each basis function maps uniquely to a dirac delta in frequency space. This allows for power/energy symmetries between time and frequency domain representations of a signal and allows for more tenable frequency filtering, e.g. removing 60/50Hz noise or isolating a particular frequency band.
A haar wavelet or modified square wave basis may provide you with a simple orthonormal signal basis, but each wavelet has a frequency representation with infinite support in frequency space. This is (a) untenable and (b) eliminates the possibility of frequency-specific filtering. Wavelet analysis is more useful in specific cases.
The sin and cos basis is very convenient. Even more convenient is the e^(i.n.t) basis...
This, by the way, was perhaps the most mindblowing thing I learned in my advanced linear algebra course. I had already learned about applied Fourier Transforms and the FFT, but seeing the same tools we had used to prove things about Cartesian bases could be applied to something so unexpected, with such powerful results, was incredible.
The FFT is still magic though :)
"Complex exponentials are the eigenfunctions of linear systems."
This means that, if you put a complex exponential in, you get the same complex exponential out, scaled by some constant, whose magnitude we call the "frequency response". No other functional form has this property.
They are eigenfunctions, so they are orthogonal. We can represent any function in L2 using a weighted sum of them.
In the first case, when you're solving a differential equation like
f(t) = - C^2 * d^2/dt^2 f(t)
A*cos(Cx+phi)
A*cos(Cx) + B*sin(Cx)
A*e^(±iCx)
In the second case, we use complex numbers to keep track of something. Like when you're doing a Fourier transform, you break down a signal into its different frequency components. However, a signal made up of a 500hz sine wave is going to be shaped differently from a signal made of a 500hz cosine wave, so we have to keep track of those separately. Conveniently, e^(ix) = cos(x) + i sin(x), so we use a complex number to represent the presence of a 500hz cosine wave (real part) and a 500hz sine wave (complex part) while keeping our equations simple. That's sort of a practical overview; a deeper understanding can come from learning linear and abstract algebra.
Another question worth at least cursory mention: what is so great about sine waves (or complex exponents)? Why are they better than, say, step functions for decomposition of many physical phenomena? Even a couple of quick examples would be helpful (drop a stone into a lake; what shape are the waves? extra credit -- why?).
Above may sound harsh, which was not my intent; I just think expanding a little can give a much better view of the algorithm and its applications.