\textbf{Taylor series of trigonometric functions}

Sine and cosine have the following Maclaurin series (which are Taylor series around \[x_0 = 0)\]: 

,,\qquad \sin(x) = \sum_{n=0}^\infty\frac{(-1)^n}{(2n + 1)!}x^{2n + 1} 
,,\qquad \cos(x) = \sum_{n=0}^\infty \frac{(-1)^n}{(2n)!}x^{2n}

This is the same as writing:

,,\qquad \sin(x) = x -\frac{x^3}{3!} + \frac{x^5}{5!} - \frac{x^7}{7!} + \frac{x^9}{9!} -\frac{x^{11}}{11!} +\dots
,,\qquad \cos(x) = 1 - \frac{x^2}{2!} + \frac{x^4}{4!} - \frac{x^6}{6!} + \frac{x^8}{8!} - \frac{x^{10}}{10}+\dots

By summing the Maclaurin series of  \[\cos(x)\] and \[i\sin(x)\] it's possible to prove Euler's formula \[e^{ix} = \cos(x) + i\sin(x)\]

,,\qquad \cos(x) + i\sin(x) = 1 + ix + \frac{(ix)^2}{2!} + \frac{(ix)^3}{3!} + \frac{(ix)^4}{4!} + \dots = e^{ix}

Maclaurin series can be used to approximate trigonometric functions to an arbitrary degree
For small angles (i.e. \[\alpha\approx 0\]), good approximations (often used in physics) are:

,,\qquad \sin(\alpha)\approx x
,,\qquad \cos(\alpha)\approx 1

Which are just Maclaurin series with only 1 term.