$\sin x$ approximates $x$ for small angles In physics, particularly in waves, we make use of the fact that for small angles (less than $\pi/12$-ish), the sine function value of an angle is pretty close to the value of the angle itself (in radians of course). Can anyone give a mathematical explanation for why this is?

Hint
Any mathematical function can be, at least locally, approximated by so called Taylor or Mc Laurin expansions.  
To make it as simple as possible, the tangent to a curve is, at the point where it is defined, a local approximation of the curve.
So, write the equation of the tangent to the curve $y=sin(x)$ at $x=0$ and you will obtain, for the tangent line,  $y = 0 + (x-0) = x$. So, close to $x=0$, $sin(x)$ is close to $x$.  
Uisng the same approach, you could show by yourself that, close to $x=0$, $e^x$ is close to $1+x$, that $log(1+x)$ is close to $x$ and so on. For sure, the approximations can be made better and better at the price of more terms. 
