How to calculate the sine manually, without any rules, calculator or anything else?
You changed very quickly in a comment to MrFatzo's answer to "how do computers calculate $\sin(x)$?", so I'm going to infer that what you're actually trying to ask is:
How does one calculate sines from scratch, without taking anyone's word for the correctness of tables or other magic values that go into the calculation?
I'm aware of two methods:
The ancients reckoned sines in degrees rather than radians. They created tables of sine values (actually chord values, in really ancient times, but that more or less amounts to the same problem) by starting with $\sin(0^\circ)=0$, $\sin(90^\circ)=1$ and then using known formulas for $\sin(v/2)$ to find sines of progressively smaller angles than $90^\circ$, and then formulas for $\sin(v+u)$ to find sines of sums of these smaller angles. That way they could eventually fill out their entire table.
In this method calculating a single sine from scratch is not really anything you do -- it's not very much less work than creating the entire table, which is to say: years and years of painstaking manual calculations.
See How to evaluate trigonometric functions by pen and paper? for a bit more detail.
In more modern times -- that is, roughly after the development of calculus -- we prefer our sines in radians. Then the gold standard for what the value of a sine should be is the power series: $$ \sin x = x - \frac16 x^3 + \frac1{120} x^5 - \cdots + \frac{(-1)^n}{(2n+1)!} x^{2n+1} + \cdots $$ This series converges quite fast when $x$ is not larger than a handful of radians, and it is simple to estimate the convergence as you go along (once $2n>x$, the limit will be strictly between any two successive partial sums), so that lets you compute single sines from scratch to any precision you desire.
The power series is still kind of slow even for computers, if you want to compute millions of sines. So in practice computers and calculators use various combinations of clever interpolation methods and tables that are built into the hardware. The tables themselves were ultimately constructed using the power series methods.
I'm not sure what can you do "manually", but maybe try using a taylor approximation?
For example, you can calculate $x-\frac{x^3}{6}$
Use the old-fashioned method: draw a really big circle, add the angle you wish to calculate, and measure.