I thought I understood the definition of transcendental numbers, but recently I was thinking about infinite series and a question was raised that I couldn't answer.
If we write a finite number of terms for the Taylor series expansion of $\sin(x)$ centered around $0$, then assuming that a finite number of terms could approximate sine past $\pi$, wouldn't this make one of the roots to the polynomial $\pi$?
I may just be misunderstanding something about transcendental numbers or forgetting an important part of the rules of series, but if this is the case, could someone point out my mistake?
The truncated Taylor polynomial for $\sin(x)$ still never actually equals $\sin(x)$ except at zero. So the polynomials will have roots as close to $\pi$ as you want, but never equal.