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.
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 $\pm 1$.