algebra precalculus - Approximation of the Sine function near $0$

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?

Answer

To see that $\sin(x) \approx x$ for small $x$ all you have to do (without using the Taylor series) is look at the graph:

You can see that $\sin x = x$ when $x = 0$, and since the gradient of the graph is approximately 1 for $-0.5

$\cos x \approx 1-\frac{x^2}{2}$

$\tan x \approx x$