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I was hoping someone could give me a flow chart or high-level map connecting all of the definitions of the sine function, with some of the reasons why we care next to each. I've tried this but I'm not able to fill in the details to my satisfaction. Here are a few definitions I know, and it's clear how some are connected to each other. There is definitely many more I am missing.

The first definition of $\sin\theta$ is usually as the angle determined in a right triangle by $$\sin^{-1}:[-1,1]\to[0,2\pi] \ , \ \frac{\text{opp}}{\text{adj}}\mapsto \sin^{-1}\left(\frac{\text{opp}}{\text{adj}}\right)$$

Or to be the $y$-coordinate of a point on the unit circle. . That is $$\sin:[0,2\pi]\to[-1,1] \ , \ \theta\mapsto \sin\theta$$

There is the continued fraction definition $$\sin:\mathbb{R}\to\mathbb{R} , \ , x\mapsto \sin x:=\dfrac{x}{1+\dfrac{x^2}{2\cdot 3-x^2+\dfrac{2\cdot 3x^2}{4\cdot 5-x^2+\dfrac{4\cdot 5x^2}{6\cdot 7-x^2+\ddots}}}}$$

There's also the power series definition $$\sin:\mathbb{R}\to\mathbb{R}\ , \ x\mapsto \sum_{n=0}^\infty\frac{(-1)^n}{(2n+1)!}x^{2n+1}$$ and the extension to $\mathbb{C}$ by power series $$\sin:\mathbb{C}\to\mathbb{C}\ , \ z\mapsto \sum_{n=0}^\infty\frac{(-1)^n}{(2n+1)!}z^{2n+1}$$ or Euler's formula $$\frac{e^{iz}-e^{-iz}}{2i}$$ or as the imaginary part divided by the modulus of a complex number.

Thomas Andrews
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user162520
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    To that I'd add the differential equations definition and the definition as the inverse of the inverse sine function (which may sound backwards, except that $\sin^{-1}x$ can be defined in terms of an appropriate indefinite integral). – Semiclassical Jul 22 '14 at 03:47
  • I think you are confusing equalities with definitions. I have never seen the continued fraction used as a definition for $\sin$. :) – Thomas Andrews Jul 22 '14 at 03:52
  • you might find some of the discussion in the comments of this recent question of interest – Semiclassical Jul 22 '14 at 03:53
  • A charitable interpretation is: what descriptions of $\sin$ are (or could be) used to define the $\sin$ function, and why certain of them are better than others... @ThomasAndrews – Semiclassical Jul 22 '14 at 03:55
  • I like to define it as the imaginary part of the limit $$\lim_{n\to\infty} \left(1+\frac{ix}{n}\right)^n$$ That's roughly the same as the imaginary part of $e^{ix}$, but I prefer it to the power series definition of $e^{ix}$. – Thomas Andrews Jul 22 '14 at 04:04
  • Anything that completely characterizes the $\sin$ function is definitive, whether there is a publication that uses it as a definition or not. – DanielV Jul 22 '14 at 04:09

2 Answers2

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A rigorous way to define the sine function is to consider it as the solution to the IVP: $$ \begin{cases} y^{\prime \prime} + y = 0\\ y(0) = 0 \\ y^{\prime}(0) = 1 \end{cases} $$

Mathsource
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Some other important formulas regarding $\sin(x)$ that you didn't mention are the infinite product $$\sin(x) = x \prod_{k=1}^{\infty}\Big( 1 - \frac{x^2}{\pi^2 k^2} \Big)$$ and the partial fractions decomposition $$\frac{1}{\sin(x)^2} = \sum_{k=-\infty}^{\infty} \frac{1}{(x-\pi k)^2}, \; \; x \notin \pi \cdot \mathbb{Z},$$ although I guess the latter only characterizes $\sin(x)$ up to $\pm 1$.

user165670
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