Man, I used to tutor math -- I wish I got questions like that from my students!
Three answers (in case you haven't found one you like yet):
A set of sine and cosine waves whose frequency is a multiple of some value form an orthogonal set of vectors/functions, which means that for any given function or vector whose domain is at most the period of the lowest frequency wave, there's exactly one set of weights whose weighted sum equals the function (with certain caveats if the function isn't discrete). There are other such sets, for example the Hadamard series, so sines and cosines aren't unique in this regard.
A complex number can be treated geometrically, as a phase + magnitude (converted to a+bi using sin & cos of course). This representation has the benefit that the magnitude of the value is readily apparent, and makes certain calculations involving multiplication and exponentiation easier.
Sine waves arise naturally in differential equations, because they are the only functions which are the negation of their own second derivative. Hence they often turn up in second-order systems with negative feedback (e.g. microphone feedback is more-or-less a sine wave).
Three answers (in case you haven't found one you like yet):
A set of sine and cosine waves whose frequency is a multiple of some value form an orthogonal set of vectors/functions, which means that for any given function or vector whose domain is at most the period of the lowest frequency wave, there's exactly one set of weights whose weighted sum equals the function (with certain caveats if the function isn't discrete). There are other such sets, for example the Hadamard series, so sines and cosines aren't unique in this regard.
A complex number can be treated geometrically, as a phase + magnitude (converted to a+bi using sin & cos of course). This representation has the benefit that the magnitude of the value is readily apparent, and makes certain calculations involving multiplication and exponentiation easier.
Sine waves arise naturally in differential equations, because they are the only functions which are the negation of their own second derivative. Hence they often turn up in second-order systems with negative feedback (e.g. microphone feedback is more-or-less a sine wave).