23. Taylor Series

d.1. Taylor Polynomial Approximation

\(\underline{\sin x \quad \text{about } x=0}\)
\(\underline{\ln x \quad \text{about } x=1}\)
\(\underline{x^{1/3} \quad \text{about } x=8}\)

a. \(\sin x\) about \(x=0\)

The Maclaurin series for \(\sin x\) is \[ \sin x=x-\dfrac{x^3}{3!}+\dfrac{x^5}{5!}-\dfrac{x^7}{7!} +\dfrac{x^9}{9!}-\dfrac{x^{11}}{11!}+\cdots \] Here are the graphs of \(\sin x\) (in BLUE) with the \(1^\text{st}\) through \(15^\text{th}\) degree Maclaurin polynomial approximations (in RED):

When a calculator needs to evaluate the \(\sin\) of some angle, it shifts the angle by a multiple of \(2\pi\) until the angle is between \(-\pi\) and \(\pi\) and then uses one of these polynomials to get the number of digits needed for the display.