TITLE: Approximation of the Sine function near $0$
QUESTION [5 upvotes]: What is the reason that  for $x<0.5$, $\sin(x)\approx x$?
Are there more known properties of these kind for other trigonometry functions?

REPLY [2 votes]: Think of the geometric interpretation of $\sin\theta$ and $\theta$ (the one using the unit circle). $\sin\theta$ is the straight-line length from $(\cos\theta,\sin\theta)$ to the $x$-axis. $\theta$ is the curved length along the circle from that point to the $x$-axis. When $\theta$ is small, we're considering a small section of the circle, and a very small section of any smooth curve looks like a line.