Calculators, to my recollection, typically use special summation identities for the sine and cosine functions that take in some angle measure, sum up a bunch of things using that measure, and then pop out a measurement.
It's sort of like how you can approximate the exponential function $e^x$ through its power series, which is the infinite sum below. (I know, you said not to get into Taylor series or anything of the like. I'm not getting into the derivation or whatever - if you simply accept for now and take me at my word that this summation makes sense and is 100% valid, it might be easier for now.)
$$e^x = 1 + \frac{x}{1!} + \frac{x^2}{2!} + \frac{x^3}{3!} + \frac{x^4}{4!} + \frac{x^5}{5!} + ...$$
Imagine sort of stopping that summation at the 10th, or 20th, or 1000th, or 10,000th term - it won't exactly be $e^x$ but it will be an approximation. Not exact, but "close enough" - fractions of a percent off, depending on how far you go.
Lots of such identities exist for various functions - some that are crazy and wacky to look at when taken at first glance, but ones that converge very quickly to "close enough" to the actual value after only a few terms. For example, we can define these summations called "Taylor series" for functions. Their derivation involves the knowledge of calculus, but they have the same core idea as the summation above for $e^x$: take an infinite sum, truncate it at some point, and you have an approximation for the value, which is more accurate the later you truncate.
The Taylor series for sine/cosine and probably the simplest technique for approximation a calculator would use:
$$cos(x) = 1 - \frac{x^2}{2!} + \frac{x^4}{4!}- \frac{x^6}{6!} + \frac{x^8}{8!} - \frac{x^{10}}{10!} + ...$$
$$sin(x) = \frac{x}{1!} - \frac{x^3}{3!} + \frac{x^5}{5!} - \frac{x^7}{7!} + \frac{x^9}{9!} - ...$$
A sort of "refinement" of this idea - I guess anyhow, it's a bit above my head, but I imagine it has the same core principle - is used in some calculators called the "CORDIC algorithm." It's not quite just some basic summation like above, but it has the same core idea. You can probably read up on it here but it seems pretty high-level so don't be surprised - https://ieeexplore.ieee.org/document/7453811
In short, how calculators typically find cosine and sine of functions are by taking known series or algorithms, derived from calculus or even higher math, and using approximation techniques to try and get just "close enough."
The underlying details of why these methods make sense is probably a bit above your head for the time being - I'm not saying it to be rude, but you need a foundation in calculus and derivatives at least to understand the concept of Taylor series, and they're probably some of the more "basic" approximation techniques for stuff of this sort.
