Apparently TI calculators use the CORIDC algorithm which involves rotation on a complex plane using complex numbers.
COordinate Rotation DIgital Computer
I would think that at least some calculators use (or used) the Taylor Series for the functions.
They would be something a calculator can do fairly easily, as opposed to the sine function itself. Taylor Series use polynomials.
That is more likely something you would see as a mathematics/physics student at the undergraduate level.
You would learn about the sine function being an ‘odd’ function and the cosine function being an ‘even’ function.
Each is an alternating series that starts with a positive term.
If you use more terms, you get more accuracy, but a calculator displays a limited number of terms. So a fairly small number of terms in the Taylor Series will give you a decent approximation for many things.
Also, these Taylor Series are more accurate with smaller values of x using less terms. If you use x = 0, they’re exactly right using only the first term.