Does $\sin(x)$ and $\cos(x)$ have representations as hypergeometric series? Through wolfram and wiki, I've learnt that these elementary functions have a representation as hypergeometric series:
$$_2F_1(\color{red}{1,1;2;-x})=\ln(x+1)$$
$$_2F_1(\color{red}{\color{red}{\frac{1}{2},\frac{1}{2};\frac{3}{2};x^2})}=\arcsin(x)$$
$$_2F_1(\color{red}{1,1;1;x})= \dfrac{1}{1-x}$$
I have never seen the hypergeometric series representation for $\sin(x)$ and $\cos(x)$. I wonder if trigonometric and hyperbolic functions have hypergeometric series' representation as those three functions? Do you know any table from mathematical handbook or references that list these special values of the Gauss hypergeometric function?
 A: Any sufficiently well-behaved function has a unique power series representation at a given point (such as the origin). The theory about that is known as Complex Variable theory and is one of the spectacular jewels of mathematics. I strongly recommend taking a basic course in it.
Unfortunately, maths is such a vast field that the hypergeometric and other "classical" functions have got squeezed out of most undergraduate math courses. There are relatively few modern textbooks on them. The old classic is Whittaker and Watson (A course in Modern Analysis - but it is far from modern in its flavour).
If you really want to look up details the NIST Handbook of Mathematical Functions is a great (but large and fairly expensive) reference book. Chapter 15, p386 gives plenty of examples of elementary functions that can be written as special cases of the hypergeometric function.
But not all "elementary" functions can be written as hypergeometric functions (and anyway "elementary" is not a preciselyy defined term). To go into this properly you have to look at the associated differential equations. But I strongly recommend against racing ahead until you have done a little more of the undergraduate syllabus.
