what is sine of a real number I never understand what the trigonometric function sine is..
We had a table that has values of sine for different angles, we by hearted it and applied to some problems and there ends the matter. Till then, sine function is related to triangles, angles.
Then comes the graph. We have been told that the figure below is the graph of the function sine. This function takes angles and gives numbers between $-1$ and $1$ and we have been told that it is a continuous function as it is clear from the graph.

Then comes taylor expansion of sine and we have $$\sin (x)=x-\frac{x^3}{3!}+\cdots$$
I know that for any infinitely differentiable function, we have taylor expansion. But how do we define differentiability of the function sine?
We define differentiability of a function from real numbers to real numbers..
But sine is a function that takes angles and gives real numbers..
Then how do we define differentiability of such a function? are we saying the real number $1$ is the degree 1? 
I am confused.. Help me..
Above content is a copy paste of a mail i received from my friend, a 1 st year undergraduate. I could answer some things vaguely i am not happy with my own answers. So, I am posting it here.. Help us (me and my friend) to understand sine function in a better way.

What is an angle and what isn't is just our interpretation of a number. Just like "3" in our equation can mean "3 apples" or "3 rotations", it just so happens that trigonometric function are useful when your number represents an angle. So... $\sin \pi$ is just a function of a real number. That $\pi$ may represent an angle ($180^\circ$ in this example), or it might represent a half-period of an oscillatory motion in time, or some factor in some formula that has nothing to do with angles - for instance, in statistics. Similar to the fact that $\pi$ appears in formulas that have nothing to do with a circle.
My point is... all functions take pure numbers (the domain may vary: analytical functions take any complex number as their arguments). No units are allowed. Unless you are dealing with abstract math, the numbers we put in these function usually mean something for us, but it depends on the context and it doesn't affect the way the function works, it's just nice for us to keep track what we're calculating. So, your assumptions that trigonometric function only work on angles, or that angles aren't real numbers, are wrong.
