The basic functions of trigonometry, $\sin$ and $\cos$, are ubiquitous in mathematics. They were originally conceived from geometry, and so it's not surprising that they consistently show up in elementary geometry contexts. Many other appearances of these functions I find shocking, though. We have of course the famous result, $$e^{ix} = \cos x + i\sin x,$$ which connects them to exponentiation. They also appear in relation to other famous functions, such as in the formulas $$\Gamma(z)\Gamma(1-z)=\frac{\pi}{\sin \pi z},$$ $$\zeta(s) = 2^s \pi^{s-1}\sin\left( \frac{\pi s}{2}\right)\Gamma(1-s)\zeta(1-s). \tag{1}$$ I find these particularly surprising because the Riemann zeta function is closely tied to prime numbers, for example in the formula $$\zeta(s) = \prod_{p\ \text{prime}} \frac{1}{1-1/p^s} \tag{2}.$$ If I had never heard of trigonometry, and you defined $\sin$ and $\cos$ for me in terms of the unit circle (ignoring non-real values of $s$), and then told me that the expressions in $(1)$ and $(2)$ are equivalent, I'd probably call you a lying idiot!
Many proofs of the solution to the Basel problem, that $\zeta(2) = \pi^2/6$, depend heavily on using properties of $\sin$ and $\cos$, which I also find remarkable. The problem of evaluating the sum $$\zeta(2) = \sum_{n=1}^{\infty} \frac{1}{n^2}$$ seems to have very little to do with $\sin$ and $\cos$, ratios of lengths of triangles, points on the unit circle, and so forth. Even just the Taylor series $$\sin x = x – \frac{x^3}{3!} + \frac{x^5}{5!} – \frac{x^7}{7!} + \cdots, $$ $$\cos x = 1 – \frac{x^2}{2!} + \frac{x^4}{4!} – \frac{x^6}{6!} + \cdots,$$ are shockingly simple, and connect these functions to problems in analysis (and from there to problems in number theory and other fields).
I could prattle on for much longer about other uses of $\sin$ and $\cos$, such as in Fourier Analysis, integration, differential equations, analytic number theory, and even (in subtler ways) in proofs of results such as the fundamental theorem of algebra.
How can we intuitively understand this? That functions considered thousands of years ago have shown up again and again in the mathematics of the past few hundred years, in fields completely unrelated to their original use and totally foreign to the people who created them, I find absolutely astonishing. Why have these ancient functions continued to be amazingly useful in mathematics, with essentially no changes being made to their original definitions? What fundamental properties of $\sin$ and $\cos$ make them pervade modern mathematics?
Best Answer
Too long for a comment...
Consider that this may be, at least in part, be due to a historical bias: since trigonometric functions were available since the earliest days of mathematics, mathematicians tried to frame their results in familiar terms instead of in other less popular functions. This effect compounds itself, as more and more results are found that involve trigonometric functions.
So perhaps it's not that sin and cos have fundamental properties that make them pervade modern mathematics, but rather that the mathematics we have developed as humans is one which is historically biased towards geometrical descriptions. In another world where drum acoustics is paramount, perhaps expanding functions in Fourier-Bessel Series is the norm.