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).
you can also show up to public lectures given by visiting math profs and afterwards ask them whatever theoretical background questions you want so long as it's not total spoon feeding
campus walls are also covered in tutor posters for hire and many of them graduate level
I am a mathematics professor. Where do you live, how is this advertised, what kind of audience does this attract, and what sort of questions get asked?
Here they mainly talk philosophy and crazy advanced, graduate level mathematics that are way beyond my comprehension and often there are industry programmers, visiting professors on vacation, math self taught geniuses who smoke a pipe with huge unkept beards that look insane, students and even this anarchist group that shows up sometimes to talk game theory.
A few Math dept profs hang out on Sundays here too where all the public chess boards are set up and are fully approachable to answer questions as long as they aren't engrossed in a game.
Clicking on the Events page for the UBC Math dept they always have visiting Math profs give free seminars to anybody who wants to show up, and it's easy to get to the university. Every month at least 5 seminars there's one coming up by a visiting prof from UC Berkeley on Lattice Poisson AKSZ Theory, a bunch of discrete math seminars, and 2 seminars today on chemical distances and shape theorems in percolation models with long-range correlations, and retractions of representation varieties of nilpotent groups.
These guys stick around afterwards and are fully approachable I would talk to them all the time about offtopic theory and went to the student bar with a few of them and other students for a few hours.
The universe likes sine waves. If you hang a weight from a spring and give it a yank, it will bob up and down according to a sine wave. Lots of resonating and oscillating things are also governed by sine waves. So scientists and engineers are forced by circumstance to learn all about sine waves.
What I really need is a decent tutor who is willing to help me specifically with the issues I have but I've sought in vain for one who is willing to free-wheel it with me instead of just copying out of a textbook. I tried one briefly and he came up with a cop-out shortly after our first lesson, because I don't think he liked the unconventional questions I was asking, like "Why and/or how are sine waves relevant to non-geometric data? The only definition I can find of a sine casts it in strictly trigonometric terms, so how is it applicable to non-trigonometric data? Is everything encoded into a representation of a triangle before these calculations are applied?" Heh, that one made him pretty annoyed and he didn't really have a good answer.
I would love to increase my background in statistics and comp sci theory (which is basic but imo sufficient, and I seem to have a better grounding than most of the CS grads I've worked with), but I don't really know of a good option to receive that training. If someone wrote tutorials for graduate-level math from the bottom principles up like they write out tutorials on PHP or whatever, I'd be all over it. I really want to increase my formal mathematical literacy.