Taylor series and its relation to sine I recently read, one of the most inspiring pieces of literature I've seen, Lockhart's Lament. And now I find myself constantly doing math for fun in my head with imaginary perfect shapes. One such puzzle got me thinking... (Side note: so thankful to SE math for letting me discover the literary gem.)
What is the relationship between sine and the taylor series? I read that most programs use the Taylor series as the basis for their algorithms for calculating sine. 
The link and information seem legitimate, even has a nice graph showing the similarities between the two functions. 
I also read this tidbit which suggests something which is seemingly quite different. What is the relationship between the two things going on here, and while we are at it, what is the actual function of sine, since the Taylor series only approximates it.
(Additional side note: bonus points for my dedication to math, I typed this on my mobile.)
bonus question: the tags suggest Taylor expansion and not Taylor series, is there a nuance in the difference or just a cute artifact of the English language?
 A: The Taylor series is not an approximation of sine.  We say that
$$
\sin(x)=x-\frac1{3!}x^3+\frac1{5!}x^5-\cdots
$$
with the same justification under which we say
$$
\pi=3.1415926535897\dots
$$
The key lies in the ellipsis, in the bit we leave out. If you stop the representation short (at the third term or $14^{th}$ decimal point), what you have is merely an approximation.  However, taken as a whole, the left and right sides of the equation produce the same thing.
Now, pi just the number equal to 
$
3.1415926535897\dots
$
? Yes, this is a thing that happens to be true for pi, but what makes pi pi is its definition as the ratio of circumference to diameter.
Similarly, the $\sin(x)$ is defined as the opposite leg to hypotenuse ratio of a right triangle with angle $x$ in radians. However, $\sin(x)$ is indeed equal to its Taylor series.
And yes, Taylor series and Taylor expansion mean the same thing in most situations, though expansion is sometimes used to refer to the finite approximation.
