I'm just going to answer the question "What about tangent, or other functions", since the rest seems to have been fairly well handled.
All sounds that we hear as having a definite pitch or note can be represented by a periodic function. As I wrote in my comment, any repeated shape represents a periodic function. Most periodic functions, both in the real world and in theory, are pretty complicated, at least mathematically.
If we want to do math on the sounds we are hearing, we will have to get a grip on these periodic functions. What would really help is a way to simplify them. As discussed, Fourier analysis lets us do exactly that. We can take a complicated periodic function with some really annoying math and break it down into simpler periodic functions where the math is much easier.
The simplest periodic functions are sine and cosine, and they are virtually the same thing and they are closely related. Tangent is a very famous periodic function and is quite helpful in basic trigonometry (literally the study of triangles). Tangent has many other uses, but most people first see the tangent function as a way to analyze triangles.
Even though we usually learn about sine, cosine, and tangent at the same time, the tangent function is actually different from sine and cosine in some important ways. It is not continuous, which means you couldn't draw very much of it without picking up your pencil and putting it down again. It's even worse than that, because it has an infinite number of discontinuities (places where you have to pick up the pencil). Note that not only is this a stark contrast with sine and cosine, but also with periodic functions that model sound waves. Sound waves are continuous functions if we plot them out.
So if we want to take a periodic function and break it into simple pieces, we don't want to break it into tangent pieces. Tangent will not help us, it's really more likely to make things worse. When Fourier (the mathematician who invented Fourier analysis) was trying to take apart periodic functions, he was looking for a way to turn them into sines and cosines, not any other periodic function (and there are many).
I could write several pages on why sine and cosine are not merely simple periodic functions, there are actually the simplest possible ones, but I don't think this is the best place for that. Let me briefly say, however, that a circle is perhaps the simplest shape, and if you follow a point as it goes around a circle, its vertical movement traces out a sine wave at the same time that its horizontal movement traces out a cosine (this depends on where on the circle the point starts, but one will be sine and the other cosine, or at least one will be one quarter cycle behind the other in phase).
Below is a video of that, and hopefully this will give you a way to see how the sine wave (or cosine, same thing, really) is the simplest periodic function, and therefore the one we want to break more complicated functions into, and also the one that to our ears appears to be the most basic if we turn it into sound.
See also the Physics version of this question with a similar answer: https://physics.stackexchange.com/questions/352754/why-cosine-and-sine-functions-are-used-while-representing-a-signal-or-a-wave