Calculating $\pi$ using $\pi=\lim_{n\to\infty}n\sin\frac{180^\circ}{n}$ How can the fact that
$$\pi=\displaystyle\lim_{n\to\infty}n\sin\frac{180^\circ}{n}$$
be useful when calculating $\pi$? I mean, $180^\circ =\pi$, isn't that a circular reasoning?
I got the idea that Lagrange interpolation might help, since every argument is divided by the corresponding power of $\pi$, therefore avoiding the circularity when we choose to interpolate
$$0\to 0,\, \frac{\pi}{2}\to 1,\, \pi\to 0,\, \frac{3\pi}{2}\to -1,\, 2\pi\to 0.$$
This interpolation yields
$$\sin x\approx \dfrac{8x^3}{3\pi ^3}-\dfrac{8x^2}{\pi ^2}+\dfrac{16x}{3\pi}.$$
But this is problematic since it's a polynomial and its behavior at $\infty$ is very different from $\sin$ at $\infty$, so that can't be used. Using
$$\sin x=\displaystyle\sum_{n=0}^\infty \dfrac{(-1)^n\, x^{2n+1}}{(2n+1)!}$$
or
$$\sin x=x\displaystyle\prod_{n=1}^\infty \left(1-\dfrac{x^2}{n^2\pi^2}\right)$$
doesn't help, since not every $x$ is divided by corresponding power of $\pi$; using such series or products to calculate $\pi$ would be circular. So, how can the formula in the question be used to calculate $\pi$?
 A: One way of using this fact is to stick to a certain subset of $\mathbb N$, namely $n = 2^k$, and evaluate $\sin\frac{\pi}{2^{k+1}}$ in terms of $\sin\frac{\pi}{2^k}$. This essentially is what Viete did to arrive to his formula.
Notice that you don't need to know $\pi$ to compute $\sin \frac{\pi}{4}$.
Another way is to interpret it in purely geometrical way: divide the half circle into $n$ congruent arcs, construct a corresponding sine, and replicate it $n$ times to get a geometric approximation of $\pi$.
A: I'm not prepared to touch on the bulk of your post, but with regard to the circularity of reasoning you seem primarily concerned about, I would beg to differ. Everything else you posted on seems to be tied to examples for which your concern holds consequences, rather than as separate questions in their own right.
Degrees as a unit of measure for angles predate the radian. The trigonometric functions are defined on angles, and not on any particular unit of measure, thus they can be measured in any type of unit we wish.
Trigonometric functions have been computed for centuries in degrees.  No, in fact thinking of trigonometric functions in terms of radians rather than more abstractly in terms of angles, in my view, IS where the circularity of reasoning is taking place. This is because it puts the radian before the function, rather than as the result of a need for a special unit in the treatment of trigonometric functions in calculus. Perhaps this perspective is too respecting of history to hold in mathematics, but I like to take math in the chronological order it was discovered; it does tend to resolve a lot of the circularity.
There is no circularity in defining $\sin(x)$ when $x$ is in degrees.  The only time there is a conflict is when $x$ also exists outside of the argument of the function - such as $x\sin(x)$, clearly the unit changes the solution. Or when by way of some sort of transformation (such as differentiation) a constant is brought out of the argument - if $x$ is in radians then $\frac{d}{dx}\sin(x)=\cos(x)$ but if $x$ is in grads then $\frac{d}{dx}\sin(x) = \frac{\pi}{200}\cos(x)$.  Also of note are the polynomial representations of trigonometric functions, approximates or otherwise, but this is because they are built on differentiation. The list goes on, I'm sure. This isnt exhaustive.  Im only suggesting that if an expression is somehow derived from a trigonometric function, especially using calculus, then care needs to be taken.
I could prove that $\frac{d}{dx}\sin(x) = \frac{\pi}{180}\cos(x)$ if I wanted to use only degrees.  The math is perfectly valid and perfectly rigorous, but if you were just blindly going through the motions you might not see why. If you thought only in terms of radians you'd probably call it wrong. But that is what the identity would look like if we never created the radian. Thus many of those expressions you wrote would contain constants involving ratios of pi and 180, and it would just get messy. That's a part of the reason for why the radian was inventned and standardized as a unit of measure for angle: it eliminates those constants. And yes, it was invented, not discovered. We chose it specifically because it plays well with calculus, but the units of angle measure are completely arbitrary.
Im sure this isnt news to you. I just thought it was worth pointing out the natural evolution of the math to resolve the apparent contradiction.  The contradiction only exists to you because, if I may speculate, you view the function not only as a function requiring a radian argument, but perhaps even as a function only computable with a Taylor (or similar) series? This is, of course, problematic. Not only does it imply circularity but it fails to appreciate history and, frankly, limits your own ability to abstract angle.
