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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?

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  • $\begingroup$ Since the answers will be highly opinion based, I am going to convert this to a community wiki question. $\endgroup$ – robjohn Oct 4 '14 at 7:36
  • $\begingroup$ Perhaps this post might interest you as well. $\endgroup$ – Lucian Oct 4 '14 at 8:22
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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.

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  • $\begingroup$ but the fact that drum acoustic is not paramount but geometry, may it reveal something about nature, no? $\endgroup$ – Ooker Sep 26 '17 at 13:18
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That's easy. $\sin$ and $\cos$ are solutions to the differential equation $\frac{d^2y}{dx^2}=-y$. What's more, they're a basis for the solutions, and a simple one at that (having value and derivative at the origin equal to 0 and 1).

Since that's such a simple equation, it's no surprise they appear often - much like their sibling $\exp$, the solution to $\frac{dy}{dx}=y$.

The equation represents a restoring force, and this is why they are so important in things that repeat, such as circles (and of course, the numerous physical applications, such as harmonic oscillators, waves etc.) This is perhaps why they are more common the hyperbolic functions - which solve $\frac{d^2y}{dx^2}=y$, but don't add much over the simpler exponential.

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All of mathematics is beautifully integrated. If there were no consistency in this one whole tree of so many branches, it would have fallen apart long ago.

Sin/cos are circular function relations that are so fundamental to rotation of a line segment or vector and must essential for any periodic event or a phenomenon.

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"Mathematics is the art of giving the same name to different things", Poincare.

I am sure sine and cosine are just a few of the many examples that one could give for mathematical objects being used in contexts different than the ones they originated in.

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From my perspective as an engineer, the development

  1. Differential and Integral Calculus
  2. Differential and Integral Equations
  3. Fourier Analysis

Here 3 hopped in and showed that with trigonometric functions we can rewrite differential and integral operators with monomial multiplication. As time went on people found out how useful differential equations are in just about every branch of math and science since then. And earliest most widespread tool to solve them is with Fourier Transforms which are basically built up with sin and cos functions.

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