I have recently been working with, and reading a bit about, infinitesimals and hyperreals and am currently trying to figure out how the trigonometric functions for infinitesimal inputs should behave and what they should "evaluate to". The problem goes something like this:
Let $\varepsilon$ be some infinitesimal, such that it is smaller than any positive real number but still larger than zero. Is it possible to evaluate $\sin \varepsilon$ or express it in terms of elementary functions (except for the trigonometric functions)?
This is rather non-trivial. The intuitive answer would simply be that $\sin \varepsilon$=$\varepsilon$. But in reality, this is an approximation and depending on the context, that approximation has to be made better. One could of course try to define it using a Taylor expansion:
$\displaystyle \sin \varepsilon=\sum_{n=0}^{\infty}\frac{(\varepsilon)^{2n+1}\cdot(-1)^n}{(2n+1)!}$
But this is quite unsatisfying, since we haven't really solved the issue, just expressed it differently. Using an infinite sum is "cheating". It may of course be impossible to express it as anything other than an infinite sum, similar to how you cannot express $\sin10$ as anything but $\sin 10$. Here is where I would like to ask for your thoughts on this. Do you have any good ideas regarding this and or resources where I can read about these kinds of things?
In the realm of nonstandard analysis, there is no hope for such a formula.
First of all, we should distinguish between formulae $\sin(\varepsilon) = f(\varepsilon)$ that happen to work for some particular infinitesimal $\varepsilon$, and formulae that would hold for all infinitesimals $\varepsilon$.
The former problem has many boring, trivial solutions. E.g. if you pick your favorite infinitesimal $\varepsilon_0$, you have the formula $\sin(\varepsilon_0) = f(\varepsilon_0)$ where $f$ denotes the constant function $f(x) = \varepsilon_0$.
The latter problem, where we want to have an elementary formula that works for all infinitesimals (or even just a large set of infinitesimals), does not admit such trivial solutions. Unfortunately, we'll see below the line that it admits no solutions at all.
I hope someone proves me wrong, but giving a satisfying explanation of why no such formula exists might prove difficult at your current level of mathematical understanding. I'll go ahead and sketch a proof anyway: it'll benefit others, and even if it goes in too deep for you momentarily, I'm sure that in a year or two you'll look back and conclude that you're now able to reconstruct the proof outlined here too.
Let $\mathcal{F}$ denote the set of trigonometry-free elementary functions defined on subintervals of the real line: the ones generated by rational powers, exponentials and logarithms by closing under sums, differences, products, reciprocals, and compositions.
Every $f \in \mathcal{F}$ constitutes a so-called definable function in the ordered exponential field of real numbers. We know quite a bit about the structure of this exponential field thanks to Wilkie's theorem: for example, it implies that we can always write the zero set of a definable function as a finite union of points and nondegenerate open intervals. Notice that $\sin$ has countably many zeroes on any unbounded interval, so it cannot coincide with any $f\in \mathcal{F}$ on such an interval.
Combining Wilkie's theorem with a result of Ax on the transcendence degrees of complex analytic functions, Bianconi's article "Nondefinability results for expansions of the field of real numbers by the exponential function and by the restricted sine function" gives a short proof of the stronger result that in fact no $f \in \mathcal{F}$ can coincide with $\sin$ on any nondegenerate open interval, bounded or unbounded.
Using a bit more of what became known as o-minimality, we can show that "frequent pointwise coincidences" between $\sin$ and $f\in \mathcal{F}$ do not happen either: by the o-minimality result for $\mathbb{R}_{an,\exp}$ we know that if $f \in \mathcal{F}$, then in a bounded interval $I$ around zero, the set of $x \in I$ so that $\sin(x) = f(x)$ is also a finite union of points and open intervals. We already know from the previous paragraph that these functions cannot coincide on an open interval: this leaves only a finite set of points around $0$ where $\sin$ coincides with the non-trigonometric formula $f$. Finally assuming that $f$ is standard, we can use the fact that $\sin$ and $f$ are locally almost linear to deduce that there is at most one infinitesimal $\varepsilon$ for which the equality $\sin(\varepsilon) = f(\varepsilon)$ holds.
It might seem that your question does not connect greatly with infinitesimals and nonstandard analysis at all: as Mark S. pointed out in the comments, one could easily ask related questions about appreciable angles, like $\sin(10^\circ)$ or $\sin(20^\circ)$ (see also Niven's theorem). The argument didn't invoke anything specific to nonstandard analysis until the very last step: it just explained why the sine function does not coincide with non-trigonometric elementary functions in general, and relied on the fact that the infinitesimals of nonstandard analysis behave very similarly to ordinary real numbers (see also Mikhail Katz's answer regarding the overspill principle).
However, we have another formulation of calculus with infinitesimal quantities, which is very different from nonstandard analysis, called smooth infinitesimal analysis (see also synthetic differential geometry).
Smooth infinitesimal analysis introduces so-called nilpotent infinitesimals, small quantities $\varepsilon$ that satisfy $\varepsilon^n = 0$ for some $n$. In this setting, we do have the equality $\sin(\varepsilon) = \varepsilon$ for all nilsquare infinitesimal quantities (similar to dual numbers which you might have seen already).
If this interests you, a couple of textbooks exist, including Bell's Primer of Infinitesimal Analysis and the first part of Kock's freely available book. In my experience, all of these have a sort of deceptive simplicity: they look like accessible introductions, but end up confusing many people who venture in without significant background in nonclassical logic or categories.